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</style><table class="sidebar sidebar-collapse nomobile nowraplinks em-sidebar"><tbody><tr><th class="sidebar-title"><a href="Electromagnetism" title="Electromagnetism">Electromagnetism</a></th></tr><tr><td class="sidebar-image"></td></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Electricity" title="Electricity">Electricity</a></li>

<li><a href="Magnetism" title="Magnetism">Magnetism</a></li>

<li><a href="Optics" title="Optics">Optics</a></li>

<li><a href="History_of_electromagnetic_theory" title="History of electromagnetic theory">History</a></li>

<li><a href="Computational_electromagnetics" title="Computational electromagnetics">Computational</a></li>

<li><a href="List_of_textbooks_in_electromagnetism" title="List of textbooks in electromagnetism">Textbooks</a></li>

<li><a href="List_of_electrical_phenomena" title="List of electrical phenomena">Phenomena</a></li></ul></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Electrostatics" title="Electrostatics">Electrostatics</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Charge_density" title="Charge density">Charge density</a></li>

<li><a href="Electrical_conductor" title="Electrical conductor">Conductor</a></li>

<li><a href="Coulomb's_law" title="Coulomb's law">Coulomb law</a></li>

<li><a href="Electret" title="Electret">Electret</a></li>

<li><a href="Electric_charge" title="Electric charge">Electric charge</a></li>

<li><a href="Electric_dipole_moment" title="Electric dipole moment">Electric dipole</a></li>

<li><a href="Electric_field" title="Electric field">Electric field</a></li>

<li><a href="Electric_flux" title="Electric flux">Electric flux</a></li>

<li><a href="Electric_potential" title="Electric potential">Electric potential</a></li>

<li><a href="Electrostatic_discharge" title="Electrostatic discharge">Electrostatic discharge</a></li>

<li><a href="Electrostatic_induction" title="Electrostatic induction">Electrostatic induction</a></li>

<li><a href="Gauss's_law" title="Gauss's law">Gauss's law</a></li>

<li><a href="Insulator_(electricity)" title="Insulator (electricity)">Insulator</a></li>

<li><a href="Permittivity" title="Permittivity">Permittivity</a></li>

<li><a href="Polarization_density" title="Polarization density">Polarization</a></li>

<li><a href="Electric_potential_energy" title="Electric potential energy">Potential energy</a></li>

<li><a href="Static_electricity" title="Static electricity">Static electricity</a></li>

<li><a href="Triboelectric_effect" title="Triboelectric effect">Triboelectricity</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Magnetostatics" title="Magnetostatics">Magnetostatics</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Amp%C3%A8re's_circuital_law" title="Ampère's circuital law">Ampère's law</a></li>

<li><a href="Biot%E2%80%93Savart_law" title="Biot–Savart law">Biot–Savart law</a></li>

<li><a href="Gauss's_law_for_magnetism" title="Gauss's law for magnetism">Gauss's law for magnetism</a></li>

<li><a href="Magnetic_moment" title="Magnetic moment">Magnetic dipole</a></li>

<li><a href="Magnetic_field" title="Magnetic field">Magnetic field</a></li>

<li><a href="Magnetic_flux" title="Magnetic flux">Magnetic flux</a></li>

<li><a href="Magnetic_scalar_potential" title="Magnetic scalar potential">Magnetic scalar potential</a></li>

<li><a href="Magnetic_vector_potential" title="Magnetic vector potential">Magnetic vector potential</a></li>

<li><a href="Magnetization" title="Magnetization">Magnetization</a></li>

<li><a href="Permeability_(electromagnetism)" title="Permeability (electromagnetism)">Permeability</a></li>

<li><a href="Right-hand_rule#Electromagnetism" title="Right-hand rule">Right-hand rule</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Classical_electromagnetism" title="Classical electromagnetism">Electrodynamics</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Bremsstrahlung" title="Bremsstrahlung">Bremsstrahlung</a></li>

<li><a href="Cyclotron_radiation" title="Cyclotron radiation">Cyclotron radiation</a></li>

<li><a href="Displacement_current" title="Displacement current">Displacement current</a></li>

<li><a href="Eddy_current" title="Eddy current">Eddy current</a></li>

<li><a href="Electromagnetic_field" title="Electromagnetic field">Electromagnetic field</a></li>

<li><a href="Electromagnetic_induction" title="Electromagnetic induction">Electromagnetic induction</a></li>

<li><a href="Electromagnetic_pulse" title="Electromagnetic pulse">Electromagnetic pulse</a></li>

<li><a href="Electromagnetic_radiation" title="Electromagnetic radiation">Electromagnetic radiation</a></li>

<li><a href="Faraday's_law_of_induction" title="Faraday's law of induction">Faraday's law</a></li>

<li><a href="Jefimenko's_equations" title="Jefimenko's equations">Jefimenko equations</a></li>

<li><a href="Larmor_formula" title="Larmor formula">Larmor formula</a></li>

<li><a href="Li%C3%A9nard%E2%80%93Wiechert_potential" title="Liénard–Wiechert potential">Liénard–Wiechert potential</a></li>

<li><a href="London_equations" title="London equations">London equations</a></li>

<li><a href="Lorentz_force" title="Lorentz force">Lorentz force</a></li>

<li><a href="Maxwell's_equations" title="Maxwell's equations">Maxwell's equations</a></li>

<li><a href="Maxwell_stress_tensor" title="Maxwell stress tensor">Maxwell tensor</a></li>

<li><a href="Poynting_vector" title="Poynting vector">Poynting vector</a></li>

<li><a href="Synchrotron_radiation" title="Synchrotron radiation">Synchrotron radiation</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Electrical_network" title="Electrical network">Electrical network</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul>

<li><a href="Capacitance" title="Capacitance">Capacitance</a></li>

<li><a href="Current_density" title="Current density">Current density</a></li>

<li><a href="Direct_current" title="Direct current">Direct current</a></li>

<li><a href="Electric_current" title="Electric current">Electric current</a></li>

<li><a href="Electric_power" title="Electric power">Electric power</a></li>

<li><a href="Electrolysis" title="Electrolysis">Electrolysis</a></li>

<li><a href="Electromotive_force" title="Electromotive force">Electromotive force</a></li>

<li><a href="Electrical_impedance" title="Electrical impedance">Impedance</a></li>

<li><a href="Inductance" title="Inductance">Inductance</a></li>

<li><a href="Joule_heating" title="Joule heating">Joule heating</a></li>

<li><a href="Kirchhoff's_circuit_laws" title="Kirchhoff's circuit laws">Kirchhoff's laws</a></li>

<li><a href="Network_analysis_(electrical_circuits)" title="Network analysis (electrical circuits)">Network analysis</a></li>

<li><a href="Series_and_parallel_circuits#Parallel_circuits" title="Series and parallel circuits">Parallel circuit</a></li>

<li><a href="Electrical_resistance_and_conductance" title="Electrical resistance and conductance">Resistance</a></li>

<li><a href="Resonator#Electromagnetics" title="Resonator">Resonant cavities</a></li>

<li><a href="Series_and_parallel_circuits#Series_circuits" title="Series and parallel circuits">Series circuit</a></li>

<li><a href="Voltage" title="Voltage">Voltage</a></li>

<li><a href="Waveguide_(radio_frequency)" title="Waveguide (radio frequency)">Waveguides</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Magnetic_circuit" title="Magnetic circuit">Magnetic circuit</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="AC_motor" title="AC motor">AC motor</a></li>

<li><a href="DC_motor" title="DC motor">DC motor</a></li>

<li><a href="Electric_machine" title="Electric machine">Electric machine</a></li>

<li><a href="Electric_motor" title="Electric motor">Electric motor</a></li>

<li><a href="Gyrator%E2%80%93capacitor_model" title="Gyrator–capacitor model">Gyrator–capacitor</a></li>

<li><a href="Induction_motor" title="Induction motor">Induction motor</a></li>

<li><a href="Linear_motor" title="Linear motor">Linear motor</a></li>

<li><a href="Magnetomotive_force" title="Magnetomotive force">Magnetomotive force</a></li>

<li><a href="Permeance" title="Permeance">Permeance</a></li>

<li><a href="Magnetic_complex_reluctance" title="Magnetic complex reluctance">Reluctance (complex)</a></li>

<li><a href="Magnetic_reluctance" title="Magnetic reluctance">Reluctance (real)</a></li>

<li><a href="Rotor_(electric)" title="Rotor (electric)">Rotor</a></li>

<li><a href="Stator" title="Stator">Stator</a></li>

<li><a href="Transformer" title="Transformer">Transformer</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c"><a href="Covariant_formulation_of_classical_electromagnetism" title="Covariant formulation of classical electromagnetism">Covariant formulation</a></div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Electromagnetic_tensor" title="Electromagnetic tensor">Electromagnetic tensor</a></li>

<li><a href="Classical_electromagnetism_and_special_relativity" title="Classical electromagnetism and special relativity">Electromagnetism and special relativity</a></li>

<li><a href="Four-current" title="Four-current">Four-current</a></li>

<li><a href="Electromagnetic_four-potential" title="Electromagnetic four-potential">Four-potential</a></li>

<li><a href="Mathematical_descriptions_of_the_electromagnetic_field" title="Mathematical descriptions of the electromagnetic field">Mathematical descriptions</a></li>

<li><a href="Maxwell's_equations_in_curved_spacetime" title="Maxwell's equations in curved spacetime">Maxwell equations in curved spacetime</a></li>

<li><a href="Relativistic_electromagnetism" title="Relativistic electromagnetism">Relativistic electromagnetism</a></li>

<li><a href="Electromagnetic_stress%E2%80%93energy_tensor" title="Electromagnetic stress–energy tensor">Stress–energy tensor</a></li></ul></div></div></td>

</tr><tr><td class="sidebar-content hlist">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content hlist">

<ul><li><a href="Andr%C3%A9-Marie_Amp%C3%A8re" title="André-Marie Ampère">Ampère</a></li>

<li><a href="Charles-Augustin_de_Coulomb" title="Charles-Augustin de Coulomb">Coulomb</a></li>

<li><a href="Albert_Einstein" title="Albert Einstein">Einstein</a></li>

<li><a href="Michael_Faraday" title="Michael Faraday">Faraday</a></li>

<li><a href="Hippolyte_Fizeau" title="Hippolyte Fizeau">Fizeau</a></li>

<li><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Gauss</a></li>

<li><a href="Oliver_Heaviside" title="Oliver Heaviside">Heaviside</a></li>

<li><a href="Hermann_von_Helmholtz" title="Hermann von Helmholtz">Helmholtz</a></li>

<li><a href="Joseph_Henry" title="Joseph Henry">Henry</a></li>

<li><a href="Heinrich_Hertz" title="Heinrich Hertz">Hertz</a></li>

<li><a href="John_Hopkinson" title="John Hopkinson">Hopkinson</a></li>

<li><a href="Oleg_D._Jefimenko" title="Oleg D. Jefimenko">Jefimenko</a></li>

<li><a href="James_Prescott_Joule" title="James Prescott Joule">Joule</a></li>

<li><a href="Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li>

<li><a href="Gustav_Kirchhoff" title="Gustav Kirchhoff">Kirchhoff</a></li>

<li><a href="Joseph_Larmor" title="Joseph Larmor">Larmor</a></li>

<li><a href="Alfred-Marie_Li%C3%A9nard" title="Alfred-Marie Liénard">Liénard</a></li>

<li><a href="Hendrik_Lorentz" title="Hendrik Lorentz">Lorentz</a></li>

<li><a href="James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li>

<li><a href="Franz_Ernst_Neumann" title="Franz Ernst Neumann">Neumann</a></li>

<li><a href="Hans_Christian_%C3%98rsted" title="Hans Christian Ørsted">Ørsted</a></li>

<li><a href="Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li>

<li><a href="John_Henry_Poynting" title="John Henry Poynting">Poynting</a></li>

<li><a href="William_Ritchie_(physicist)" title="William Ritchie (physicist)">Ritchie</a></li>

<li><a href="F%C3%A9lix_Savart" title="Félix Savart">Savart</a></li>

<li><a href="George_Singer" title="George Singer">Singer</a></li>

<li><a href="Charles_Proteus_Steinmetz" title="Charles Proteus Steinmetz">Steinmetz</a></li>

<li><a href="Nikola_Tesla" title="Nikola Tesla">Tesla</a></li>

<li><a href="J._J._Thomson" title="J. J. Thomson">Thomson</a></li>

<li><a href="Alessandro_Volta" title="Alessandro Volta">Volta</a></li>

<li><a href="Wilhelm_Eduard_Weber" title="Wilhelm Eduard Weber">Weber</a></li>

<li><a href="Emil_Wiechert" title="Emil Wiechert">Wiechert</a></li></ul></div></div></td>

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<p><b>Alternating current</b> (<b>AC</b>) is an <a href="Electric_current" title="Electric current">electric current</a> that periodically reverses direction and changes its magnitude continuously with time, in contrast to <a href="Direct_current" title="Direct current">direct current</a> (DC), which flows only in one direction. Alternating current is the form in which <a href="Electric_power" title="Electric power">electric power</a> is delivered to businesses and residences, and it is the form of <a href="Electrical_energy" title="Electrical energy">electrical energy</a> that consumers typically use when they plug <a href="Kitchen_appliance" class="mw-redirect" title="Kitchen appliance">kitchen appliances</a>, <a href="Television" title="Television">televisions</a>, <a href="Fan_(machine)" title="Fan (machine)">fans</a> and <a href="Electric_lamp" class="mw-redirect" title="Electric lamp">electric lamps</a> into a <a href="Wall_socket" class="mw-redirect" title="Wall socket">wall socket</a>. The abbreviations <i>AC</i> and <i>DC</i> are often used to mean simply <i>alternating</i> and <i>direct</i>, respectively, as when they modify <i><a href="Electric_current" title="Electric current">current</a></i> or <i><a href="Voltage" title="Voltage">voltage</a></i>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>The usual <a href="Waveform" title="Waveform">waveform</a> of alternating current in most electric power circuits is a <a href="Sine_wave" title="Sine wave">sine wave</a>, whose positive half-period corresponds with positive direction of the current and vice versa (the full period is called a <i><a href="Wave_cycle" class="mw-redirect" title="Wave cycle">cycle</a></i>). "Alternating current" most commonly refers to power distribution, but a wide range of other applications are technically alternating current although it is less common to describe them by that term. In many applications, like <a href="Guitar_amplifier" title="Guitar amplifier">guitar amplifiers</a>, different waveforms are used, such as <a href="Triangle_wave" title="Triangle wave">triangular waves</a> or <a href="Square_wave_(waveform)" title="Square wave (waveform)">square waves</a>. <a href="Audio_frequency" title="Audio frequency">Audio</a> and <a href="Radio_frequency" title="Radio frequency">radio</a> signals carried on electrical wires are also examples of alternating current. These types of alternating current carry information such as sound (audio) or images (video) sometimes carried by <a href="Modulation" class="mw-redirect" title="Modulation">modulation</a> of an AC carrier signal. These currents typically alternate at higher frequencies than those used in power transmission.

</p>

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<div class="mw-heading mw-heading2"><h2 id="Transmission,_distribution,_and_domestic_power_supply">Transmission, distribution, and domestic power supply</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Electric_power_transmission" title="Electric power transmission">Electric power transmission</a> and <a href="Electric_power_distribution" title="Electric power distribution">Electric power distribution</a></div>

<p>Electrical energy is distributed as alternating current because AC <a href="Voltage" title="Voltage">voltage</a> may be increased or decreased with a <a href="Transformer" title="Transformer">transformer</a>. This allows the power to be transmitted through <a href="Power_line" class="mw-redirect" title="Power line">power lines</a> efficiently at <a href="High_voltage" title="High voltage">high voltage</a>, which reduces the energy lost as heat due to <a href="Electrical_resistance" class="mw-redirect" title="Electrical resistance">resistance</a> of the wire, and transformed to a lower, safer voltage for use. Use of a higher voltage leads to significantly more efficient transmission of power. The power losses (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\rm {w}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{\rm {w}}}</annotation>

</semantics>

</math></span><img src="./abc657948b2077aa4e0c1c2f3de7ec851c1078ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.911ex; height:2.509ex;" alt="{\displaystyle P_{\rm {w}}}" loading="lazy"></span>) in the wire are a product of the square of the current ( I ) and the <a href="Electrical_resistance" class="mw-redirect" title="Electrical resistance">resistance</a> (R) of the wire, described by the formula:

</p>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{\rm {w}}=I^{2}R\,.}</annotation>

</semantics>

</math></span><img src="./8efda57cea8199be3ce8d9667896d8963d4b2395.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.078ex; height:3.009ex;" alt="{\displaystyle P_{\rm {w}}=I^{2}R\,.}" loading="lazy"></span></dd></dl>

<p>This means that when transmitting a fixed power on a given wire, if the current is halved (i.e. the voltage is doubled), the power loss due to the wire's resistance will be reduced to one quarter.

</p><p>The power transmitted is equal to the product of the current and the voltage (assuming no phase difference); that is,

</p>

<msub>

</mrow>

</msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{\rm {t}}=IV\,.}</annotation>

</semantics>

</math></span><img src="./6fb217fe2922665f20d2077f6c2a9df94994ec37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.456ex; height:2.509ex;" alt="{\displaystyle P_{\rm {t}}=IV\,.}" loading="lazy"></span></dd></dl>

<p>Consequently, power transmitted at a higher voltage requires less loss-producing current than for the same power at a lower voltage. Power is often transmitted at hundreds of kilovolts on <a href="Transmission_tower" title="Transmission tower">pylons</a>, and transformed down to tens of kilovolts to be transmitted on lower level lines, and finally transformed down to 100 V – 240 V for domestic use.

</p>

<p>High voltages have disadvantages, such as the increased insulation required, and generally increased difficulty in their safe handling. In a <a href="Power_plant" class="mw-redirect" title="Power plant">power plant</a>, energy is generated at a convenient voltage for the design of a <a href="Electric_generator" title="Electric generator">generator</a>, and then stepped up to a high voltage for transmission. Near the loads, the transmission voltage is stepped down to the voltages used by equipment. Consumer voltages vary somewhat depending on the country and size of load, but generally motors and lighting are built to use up to a few hundred volts between phases. The voltage delivered to equipment such as lighting and motor loads is standardized, with an allowable range of voltage over which equipment is expected to operate. Standard power utilization voltages and percentage tolerance vary in the different <a href="Mains_power_systems" class="mw-redirect" title="Mains power systems">mains power systems</a> found in the world.

</p><p><a href="High-voltage_direct_current" title="High-voltage direct current">High-voltage direct-current</a> (HVDC) electric power transmission systems have become more viable as technology has provided efficient means of changing the voltage of DC power. Transmission with high voltage direct current was not feasible in the early days of <a href="Electric_power_transmission" title="Electric power transmission">electric power transmission</a>, as there was then no economically viable way to step the voltage of DC down for end user applications such as lighting incandescent bulbs.

</p><p><a href="Three-phase_electric_power" title="Three-phase electric power">Three-phase</a> electrical generation is very common. The simplest way is to use three separate coils in the generator <a href="Stator" title="Stator">stator</a>, physically offset by an angle of 120° (one-third of a complete 360° phase) to each other. Three current waveforms are produced that are equal in magnitude and 120° <a href="Out_of_phase" class="mw-redirect" title="Out of phase">out of phase</a> to each other. If coils are added opposite to these (60° spacing), they generate the same phases with reverse <a href="Electric_polarity" class="mw-redirect" title="Electric polarity">polarity</a> and so can be simply wired together. In practice, higher <i>pole orders</i> are commonly used. For example, a 12-pole machine would have 36 coils (10° spacing). The advantage is that lower rotational speeds can be used to generate the same frequency. For example, a 2-pole machine running at 3600 rpm and a 12-pole machine running at 600 rpm produce the same frequency; the lower speed is preferable for larger machines. If the load on a three-phase system is balanced equally among the phases, no current flows through the <a href="Neutral_point" class="mw-redirect" title="Neutral point">neutral point</a>. Even in the worst-case unbalanced (linear) load, the neutral current will not exceed the highest of the phase currents. Non-linear loads (e.g. the switch-mode power supplies widely used) may require an oversized neutral bus and neutral conductor in the upstream distribution panel to handle <a href="Harmonic_(electrical_power)" class="mw-redirect" title="Harmonic (electrical power)">harmonics</a>. Harmonics can cause neutral conductor current levels to exceed that of one or all phase conductors.

</p><p>For three-phase at utilization voltages a four-wire system is often used. When stepping down three-phase, a transformer with a Delta (3-wire) primary and a Star (4-wire, center-earthed) secondary is often used so there is no need for a neutral on the supply side. For smaller customers (just how small varies by country and age of the installation) only a <a href="Single-phase_electric_power" title="Single-phase electric power">single phase</a> and neutral, or two phases and neutral, are taken to the property. For larger installations, all three phases and neutral are taken to the main distribution panel. From the three-phase main panel, both single and three-phase circuits may lead off. <a href="Split-phase_electric_power" title="Split-phase electric power">Three-wire single-phase</a> systems, with a single center-tapped transformer giving two live conductors, is a common distribution scheme for residential and small commercial buildings in North America. This arrangement is sometimes incorrectly referred to as <i>two phase</i>. A similar method is used for a different reason on construction sites in the UK. Small power tools and lighting are supposed to be supplied by a local center-tapped transformer with a voltage of 55 V between each power conductor and earth. This significantly reduces the risk of <a href="Electric_shock" class="mw-redirect" title="Electric shock">electric shock</a> in the event that one of the live conductors becomes exposed through an equipment fault whilst still allowing a reasonable voltage of 110 V between the two conductors for running the tools.

</p><p>An <a href="Ground_and_neutral" title="Ground and neutral">additional wire</a>, called the bond (or earth) wire, is often connected between non-current-carrying metal enclosures and earth ground. This conductor provides protection from electric shock due to accidental contact of circuit conductors with the metal chassis of portable appliances and tools. Bonding all non-current-carrying metal parts into one complete system ensures there is always a low <a href="Electrical_impedance" title="Electrical impedance">electrical impedance</a> path to ground sufficient to carry any <a href="Fault_(power_engineering)" class="mw-redirect" title="Fault (power engineering)">fault</a> current for as long as it takes for the system to clear the fault. This low impedance path allows the maximum amount of fault current, causing the overcurrent protection device (breakers, fuses) to trip or burn out as quickly as possible, bringing the electrical system to a safe state. All bond wires are bonded to ground at the main service panel, as is the neutral/identified conductor if present.

</p>

<div class="mw-heading mw-heading2"><h2 id="AC_power_supply_frequencies">AC power supply frequencies</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Mains_electricity_by_country" title="Mains electricity by country">Mains electricity by country</a></div>

<p>The <a href="Utility_frequency" title="Utility frequency">frequency of the electrical system</a> varies by country and sometimes within a country; most electric power is generated at either 50 or <span class="nowrap">60 <a href="Hertz" title="Hertz">Hz</a></span>. Some countries have a mixture of 50 Hz and 60 Hz supplies, notably <a href="Electricity_sector_in_Japan#Transmission" title="Electricity sector in Japan">electricity power transmission in Japan</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Low_frequency">Low frequency</h3></div>

<p>A low frequency eases the design of electric motors, particularly for hoisting, crushing and rolling applications, and commutator-type <a href="Traction_motor" title="Traction motor">traction motors</a> for applications such as <a href="Railway" class="mw-redirect" title="Railway">railways</a>. However, low frequency also causes noticeable flicker in <a href="Arc_lamp" title="Arc lamp">arc lamps</a> and <a href="Incandescent_light_bulb" title="Incandescent light bulb">incandescent light bulbs</a>. The use of lower frequencies also provided the advantage of lower transmission losses, which are proportional to frequency.

</p><p>The original Niagara Falls generators were built to produce 25 Hz power, as a compromise between low frequency for traction and heavy induction motors, while still allowing incandescent lighting to operate (although with noticeable flicker). Most of the 25 Hz residential and commercial customers for Niagara Falls power were converted to 60 Hz by the late 1950s, although some 25 Hz industrial customers still existed as of the start of the 21st century. 16.7 Hz power (formerly 16 2/3 Hz) is still used in some European rail systems, such as in <a href="Austria" title="Austria">Austria</a>, <a href="Germany" title="Germany">Germany</a>, <a href="Norway" title="Norway">Norway</a>, <a href="Sweden" title="Sweden">Sweden</a> and <a href="Switzerland" title="Switzerland">Switzerland</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="High_frequency">High frequency</h3></div>

<p>Off-shore, military, textile industry, marine, aircraft, and spacecraft applications sometimes use 400 Hz, for benefits of reduced weight of apparatus or higher motor speeds. Computer <a href="Mainframe_computer" title="Mainframe computer">mainframe</a> systems were often powered by 400 Hz or 415 Hz for benefits of <a href="Ripple_(electrical)" title="Ripple (electrical)">ripple</a> reduction while using smaller internal AC to DC conversion units.

</p>

<div class="mw-heading mw-heading2"><h2 id="Effects_at_high_frequencies">Effects at high frequencies</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Skin_effect" title="Skin effect">Skin effect</a></div>

<p>A direct current flows uniformly throughout the cross-section of a homogeneous <a href="Electrical_conductivity" class="mw-redirect" title="Electrical conductivity">electrically conducting</a> wire. An alternating current of any frequency is forced away from the wire's center, toward its outer surface. This is because an alternating current (which is the result of the acceleration of <a href="Electric_charge" title="Electric charge">electric charge</a>) creates <a href="Electromagnetic_waves" class="mw-redirect" title="Electromagnetic waves">electromagnetic waves</a> (a phenomenon known as <a href="Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a>). Electric conductors are not conducive to electromagnetic waves (a <a href="Perfect_conductor" title="Perfect conductor">perfect electric conductor</a> prohibits all electromagnetic waves within its boundary), so a wire that is made of a non-perfect conductor (a conductor with finite, rather than infinite, electrical conductivity) pushes the alternating current, along with their associated electromagnetic fields, away from the wire's center. The phenomenon of alternating current being pushed away from the center of the conductor is called <a href="Skin_effect" title="Skin effect">skin effect</a>, and a direct current does not exhibit this effect, since a direct current does not create electromagnetic waves.

</p><p>At very high frequencies, the current no longer flows <i>in</i> the wire, but effectively flows <i>on</i> the surface of the wire, within a thickness of a few <a href="Skin_depth" class="mw-redirect" title="Skin depth">skin depths</a>. The skin depth is the thickness at which the current density is reduced by 63%. Even at relatively low frequencies used for power transmission (50 Hz – 60 Hz), non-uniform distribution of current still occurs in sufficiently thick <a href="Electrical_conductor" title="Electrical conductor">conductors</a>. For example, the skin depth of a copper conductor is approximately 8.57 mm at 60 Hz, so high-current conductors are usually hollow to reduce their mass and cost. This tendency of alternating current to flow predominantly in the periphery of conductors reduces the effective cross-section of the conductor. This increases the effective AC <a href="Electrical_resistance" class="mw-redirect" title="Electrical resistance">resistance</a> of the conductor since resistance is inversely proportional to the cross-sectional area. A conductor's AC resistance is higher than its DC resistance, causing a higher energy loss due to <a href="Ohmic_heating" class="mw-redirect" title="Ohmic heating">Ohmic heating</a> (also called I<sup>2</sup>R loss).

</p>

<div class="mw-heading mw-heading3"><h3 id="Techniques_for_reducing_AC_resistance">Techniques for reducing AC resistance</h3></div>

<p>For low to medium frequencies, conductors can be divided into stranded wires, each insulated from the others, with the relative positions of individual strands specially arranged within the conductor bundle. Wire constructed using this technique is called <a href="Litz_wire" title="Litz wire">Litz wire</a>. This measure helps to partially mitigate skin effect by forcing more equal current throughout the total cross section of the stranded conductors. Litz wire is used for making <a href="Quality_factor" class="mw-redirect" title="Quality factor">high-Q</a> <a href="Inductor" title="Inductor">inductors</a>, reducing losses in flexible conductors carrying very high currents at lower frequencies, and in the windings of devices carrying higher <a href="Radio_frequency" title="Radio frequency">radio frequency</a> current (up to hundreds of kilohertz), such as switch-mode <a href="Power_supply" title="Power supply">power supplies</a> and <a href="Radio_frequency" title="Radio frequency">radio frequency</a> <a href="Transformer" title="Transformer">transformers</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Techniques_for_reducing_radiation_loss">Techniques for reducing radiation loss</h3></div>

<p>As written above, an alternating current is made of <a href="Electric_charge" title="Electric charge">electric charge</a> under periodic <a href="Acceleration" title="Acceleration">acceleration</a>, which causes <a href="Electromagnetic_radiation" title="Electromagnetic radiation">radiation</a> of <a href="Electromagnetic_waves" class="mw-redirect" title="Electromagnetic waves">electromagnetic waves</a>. Energy that is radiated is lost. Depending on the frequency, different techniques are used to minimize the loss due to radiation.

</p>

<div class="mw-heading mw-heading4"><h4 id="Twisted_pairs">Twisted pairs</h4></div>

<p>At frequencies up to about 1 GHz, pairs of wires are twisted together in a cable, forming a <a href="Twisted_pair" title="Twisted pair">twisted pair</a>. This reduces losses from <a href="Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a> and <a href="Inductive_coupling" title="Inductive coupling">inductive coupling</a>. A twisted pair must be used with a <a href="Balanced_line" title="Balanced line">balanced</a> signaling system so that the two wires carry equal but opposite currents. Each wire in a twisted pair radiates a signal, but it is effectively canceled by radiation from the other wire, resulting in almost no radiation loss.

</p>

<div class="mw-heading mw-heading4"><h4 id="Coaxial_cables">Coaxial cables</h4></div>

<p><a href="Coaxial_cable" title="Coaxial cable">Coaxial cables</a> are commonly used at <a href="Audio_frequency" title="Audio frequency">audio frequencies</a> and above for convenience. A coaxial cable has a conductive wire inside a conductive tube, separated by a <a href="Dielectric" title="Dielectric">dielectric</a> layer. The current flowing on the surface of the inner conductor is equal and opposite to the current flowing on the inner surface of the outer tube. The electromagnetic field is thus completely contained within the tube, and (ideally) no energy is lost to radiation or coupling outside the tube. Coaxial cables have acceptably small losses for frequencies up to about 5 GHz. For <a href="Microwave" title="Microwave">microwave</a> frequencies greater than 5 GHz, the losses (due mainly to the dielectric separating the inner and outer tubes being a non-ideal insulator) become too large, making <a href="Waveguide_(electromagnetism)" class="mw-redirect" title="Waveguide (electromagnetism)">waveguides</a> a more efficient medium for transmitting energy. Coaxial cables often use a perforated dielectric layer to separate the inner and outer conductors in order to minimize the power dissipated by the dielectric.

</p>

<div class="mw-heading mw-heading4"><h4 id="Waveguides">Waveguides</h4></div>

<p><a href="Waveguide_(electromagnetism)" class="mw-redirect" title="Waveguide (electromagnetism)">Waveguides</a> are similar to coaxial cables, as both consist of tubes, with the biggest difference being that waveguides have no inner conductor. Waveguides can have any arbitrary cross section, but rectangular cross sections are the most common. Because waveguides do not have an inner conductor to carry a return current, waveguides cannot deliver energy by means of an <a href="Electric_current" title="Electric current">electric current</a>, but rather by means of a <i>guided</i> <a href="Electromagnetic_field" title="Electromagnetic field">electromagnetic field</a>. Although <a href="Current_density" title="Current density">surface currents</a> do flow on the inner walls of the waveguides, those surface currents do not carry power. Power is carried by the guided electromagnetic fields. The surface currents are set up by the guided electromagnetic fields and have the effect of keeping the fields inside the waveguide and preventing leakage of the fields to the space outside the waveguide. Waveguides have dimensions comparable to the <a href="Wavelength" title="Wavelength">wavelength</a> of the alternating current to be transmitted, so they are feasible only at microwave frequencies. In addition to this mechanical feasibility, <a href="Electrical_resistance" class="mw-redirect" title="Electrical resistance">electrical resistance</a> of the non-ideal metals forming the walls of the waveguide causes <a href="Dissipation" title="Dissipation">dissipation</a> of power (surface currents flowing on lossy <a href="Electrical_conductor" title="Electrical conductor">conductors</a> dissipate power). At higher frequencies, the power lost to this dissipation becomes unacceptably large.

</p>

<div class="mw-heading mw-heading4"><h4 id="Fiber_optics">Fiber optics</h4></div>

<p>At frequencies greater than 200 GHz, waveguide dimensions become impractically small, and the <a href="Ohmic_heating" class="mw-redirect" title="Ohmic heating">ohmic losses</a> in the waveguide walls become large. Instead, <a href="Fibre_optics" class="mw-redirect" title="Fibre optics">fiber optics</a>, which are a form of dielectric waveguides, can be used. For such frequencies, the concepts of voltages and currents are no longer used.

</p>

<div class="mw-heading mw-heading2"><h2 id="Formulation">Formulation</h2></div>

<p>Alternating currents are accompanied (or caused) by alternating voltages. An AC voltage <i>v</i> can be described mathematically as a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> of time by the following equation:

</p>

<msub>

</mrow>

</msub>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle v(t)=V_{\text{peak}}\sin(\omega t)}</annotation>

</semantics>

</math></span><img src="./7c05731005b2d245f908da771c57de4da5b89bcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.134ex; height:3.009ex;" alt="{\displaystyle v(t)=V_{\text{peak}}\sin(\omega t)}" loading="lazy"></span>,</dd></dl>

<p>where

</p>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{\text{peak}}}</annotation>

</semantics>

</math></span><img src="./57804b7a3ccbc7cf3a67e709854cdd7919aab732.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.921ex; height:2.843ex;" alt="{\displaystyle V_{\text{peak}}}" loading="lazy"></span> is the peak voltage (unit: <a href="Volt" title="Volt">volt</a>),</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

</semantics>

</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span> is the <a href="Angular_frequency" title="Angular frequency">angular frequency</a> (unit: <a href="Radians_per_second" class="mw-redirect" title="Radians per second">radians per second</a>). <div class="paragraphbreak" style="margin-top:0.5em"></div>The angular frequency is related to the physical frequency, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> (unit: <a href="Hertz" title="Hertz">hertz</a>), which represents the number of cycles per second, by the equation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =2\pi f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =2\pi f}</annotation>

</semantics>

</math></span><img src="./7a1bf35d395c2d52391265e4bbda0aed14f52579.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.317ex; height:2.509ex;" alt="{\displaystyle \omega =2\pi f}" loading="lazy"></span>.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t}</annotation>

</semantics>

</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span> is the time (unit: <a href="Second" title="Second">second</a>).</li></ul>

<p>The peak-to-peak value of an AC voltage is defined as the difference between its positive peak and its negative peak. Since the maximum value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(x)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \sin(x)}</annotation>

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</math></span><img src="./3a990a5545cac26c1c6821dca95d898bc80fe3a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.995ex; height:2.843ex;" alt="{\displaystyle \sin(x)}" loading="lazy"></span> is +1 and the minimum value is −1, an AC voltage swings between <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +V_{\text{peak}}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle +V_{\text{peak}}}</annotation>

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</math></span><img src="./de3811465bc7eedda488bc20e052c6734d4a47d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.73ex; height:2.843ex;" alt="{\displaystyle +V_{\text{peak}}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -V_{\text{peak}}}">

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle -V_{\text{peak}}}</annotation>

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</math></span><img src="./2094fdab248b0420bbdd07e30eb603aec5bc45a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.73ex; height:2.843ex;" alt="{\displaystyle -V_{\text{peak}}}" loading="lazy"></span>. The peak-to-peak voltage, usually written as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\text{pp}}}">

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<annotation encoding="application/x-tex">{\displaystyle V_{\text{pp}}}</annotation>

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</math></span><img src="./0adf4eef1890e2ae65472447581c6e57464a6b9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.415ex; height:2.843ex;" alt="{\displaystyle V_{\text{pp}}}" loading="lazy"></span> or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\text{P-P}}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle V_{\text{P-P}}}</annotation>

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</math></span><img src="./2c980c2388818f19bfc8f4a7c16fd38ecf870584.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.374ex; height:2.509ex;" alt="{\displaystyle V_{\text{P-P}}}" loading="lazy"></span>, is therefore <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\text{peak}}-(-V_{\text{peak}})=2V_{\text{peak}}}">

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<annotation encoding="application/x-tex">{\displaystyle V_{\text{peak}}-(-V_{\text{peak}})=2V_{\text{peak}}}</annotation>

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</math></span><img src="./f592666e37d1b4e6f0c516d24bbb8dc26726eb3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.483ex; height:3.009ex;" alt="{\displaystyle V_{\text{peak}}-(-V_{\text{peak}})=2V_{\text{peak}}}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Root_mean_square_voltage">Root mean square voltage</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="RMS_amplitude" class="mw-redirect" title="RMS amplitude">RMS amplitude</a></div>

<div role="note" class="hatnote navigation-not-searchable">For broader coverage of this topic, see <a href="Root_mean_square_voltage" class="mw-redirect" title="Root mean square voltage">Root mean square voltage</a>.</div>

<p>Below an AC waveform (with no <a href="DC_component" class="mw-redirect" title="DC component">DC component</a>) is assumed.

</p><p>The RMS voltage is the square root of the <a href="Mean_of_a_function" title="Mean of a function">mean</a> over one cycle of the square of the instantaneous voltage.

</p>

<div><ul><li>For an arbitrary periodic waveform <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(t)}">

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<annotation encoding="application/x-tex">{\displaystyle v(t)}</annotation>

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</math></span><img src="./243a0bf98a12f48552ba6a70302122d81b237b3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.777ex; height:2.843ex;" alt="{\displaystyle v(t)}" loading="lazy"></span> of period <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math></span><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span>:

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<annotation encoding="application/x-tex">{\displaystyle V_{\text{rms}}={\sqrt {{\frac {1}{T}}\int _{0}^{T}{[v(t)]^{2}dt}}}.}</annotation>

</semantics>

</math></span><img src="./ef708d0af8135ef76f3d2398403e698f8e7b98ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25.598ex; height:7.509ex;" alt="{\displaystyle V_{\text{rms}}={\sqrt {{\frac {1}{T}}\int _{0}^{T}{[v(t)]^{2}dt}}}.}" loading="lazy"></span></dd></dl></li><li>For a sinusoidal voltage:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V_{\text{rms}}&={\sqrt {{\frac {1}{T}}\int _{0}^{T}[{V_{\text{peak}}\sin(\omega t+\phi )]^{2}dt}}}\\&=V_{\text{peak}}{\sqrt {{\frac {1}{2T}}\int _{0}^{T}[{1-\cos(2\omega t+2\phi )]dt}}}\\&=V_{\text{peak}}{\sqrt {{\frac {1}{2T}}\int _{0}^{T}{dt}}}\\&={\frac {V_{\text{peak}}}{\sqrt {2}}}\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./7d702e731c7dae922caedc8378d246e417f156f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.338ex; width:44.328ex; height:29.843ex;" alt="{\displaystyle {\begin{aligned}V_{\text{rms}}&={\sqrt {{\frac {1}{T}}\int _{0}^{T}[{V_{\text{peak}}\sin(\omega t+\phi )]^{2}dt}}}\\&=V_{\text{peak}}{\sqrt {{\frac {1}{2T}}\int _{0}^{T}[{1-\cos(2\omega t+2\phi )]dt}}}\\&=V_{\text{peak}}{\sqrt {{\frac {1}{2T}}\int _{0}^{T}{dt}}}\\&={\frac {V_{\text{peak}}}{\sqrt {2}}}\end{aligned}}}" loading="lazy"></span></dd>

where the <a href="Trigonometric_identity" class="mw-redirect" title="Trigonometric identity">trigonometric identity</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin ^{2}(x)={\frac {1-\cos(2x)}{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle \sin ^{2}(x)={\frac {1-\cos(2x)}{2}}}</annotation>

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</math></span><img src="./2babf93bbda4c46a82cf91fa319b7ee98b5ef77b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.399ex; height:5.676ex;" alt="{\displaystyle \sin ^{2}(x)={\frac {1-\cos(2x)}{2}}}" loading="lazy"></span> has been used and the factor <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {2}}}">

<msqrt>

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<annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation>

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</math></span><img src="./b4afc1e27d418021bf10898eb44a7f5f315735ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}" loading="lazy"></span> is called the <a href="Crest_factor" title="Crest factor">crest factor</a>, which varies for different waveforms.</li><li>For a <a href="Triangle_wave" title="Triangle wave">triangle waveform</a> centered about zero

<msub>

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<annotation encoding="application/x-tex">{\displaystyle V_{\text{rms}}={\frac {V_{\text{peak}}}{\sqrt {3}}}.}</annotation>

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</math></span><img src="./393328da39ebe29095e07a03c32dcbbb30f6c9f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:13.752ex; height:6.509ex;" alt="{\displaystyle V_{\text{rms}}={\frac {V_{\text{peak}}}{\sqrt {3}}}.}" loading="lazy"></span></dd></dl></li><li>For a <a href="Square_wave_(waveform)" title="Square wave (waveform)">square waveform</a> centered about zero

<msub>

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<annotation encoding="application/x-tex">{\displaystyle V_{\text{rms}}=V_{\text{peak}}.}</annotation>

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</math></span><img src="./34a5df4cb1a345c1e8b1f2d3667323af9bcc2925.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.916ex; height:2.843ex;" alt="{\displaystyle V_{\text{rms}}=V_{\text{peak}}.}" loading="lazy"></span></dd></li></ul></div>

<div class="mw-heading mw-heading3"><h3 id="Power">Power</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="AC_power" title="AC power">AC power</a></div>

<p>The relationship between voltage and the power delivered is:

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<annotation encoding="application/x-tex">{\displaystyle p(t)={\frac {v^{2}(t)}{R}}}</annotation>

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</math></span><img src="./f16c2878c3406e79780c1e537a42025577028bba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; margin-left: -0.089ex; width:12.673ex; height:6.009ex;" alt="{\displaystyle p(t)={\frac {v^{2}(t)}{R}}}" loading="lazy"></span>,</dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

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</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> represents a load resistance.

</p><p>Rather than using instantaneous power, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p(t)}">

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<annotation encoding="application/x-tex">{\displaystyle p(t)}</annotation>

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</math></span><img src="./9b827c545ca1487214f0c498131228ef87718ece.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:3.908ex; height:2.843ex;" alt="{\displaystyle p(t)}" loading="lazy"></span>, it is more practical to use a time-averaged power (where the averaging is performed over any integer number of cycles). Therefore, AC voltage is often expressed as a <a href="Root_mean_square" title="Root mean square">root mean square</a> (RMS) value, written as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\text{rms}}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle V_{\text{rms}}}</annotation>

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</math></span><img src="./af4009701e886869febe6552f9a97b08cdc08066.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.249ex; height:2.509ex;" alt="{\displaystyle V_{\text{rms}}}" loading="lazy"></span>, because

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<annotation encoding="application/x-tex">{\displaystyle P_{\text{average}}={\frac {{V_{\text{rms}}}^{2}}{R}}.}</annotation>

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</math></span><img src="./45882396ab9fbb65d1e248e685b47c90b913353d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:17.048ex; height:5.843ex;" alt="{\displaystyle P_{\text{average}}={\frac {{V_{\text{rms}}}^{2}}{R}}.}" loading="lazy"></span></dd></dl>

<dl><dt>Power oscillation</dt>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}v(t)&=V_{\text{peak}}\sin(\omega t)\\i(t)&={\frac {v(t)}{R}}={\frac {V_{\text{peak}}}{R}}\sin(\omega t)\\p(t)&=v(t)i(t)={\frac {(V_{\text{peak}})^{2}}{R}}\sin ^{2}(\omega t)={\frac {(V_{\text{peak}})^{2}}{2R}}\ (1-\cos(2\omega t))\end{aligned}}}</annotation>

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</math></span><img src="./970bc0c19b0ab8da811293b6fcdcc7f755ac6eac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:61.488ex; height:15.176ex;" alt="{\displaystyle {\begin{aligned}v(t)&=V_{\text{peak}}\sin(\omega t)\\i(t)&={\frac {v(t)}{R}}={\frac {V_{\text{peak}}}{R}}\sin(\omega t)\\p(t)&=v(t)i(t)={\frac {(V_{\text{peak}})^{2}}{R}}\sin ^{2}(\omega t)={\frac {(V_{\text{peak}})^{2}}{2R}}\ (1-\cos(2\omega t))\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>For this reason, AC power's waveform becomes <a href="Rectifier#Full-wave_rectification" title="Rectifier">Full-wave rectified</a> sine, and its fundamental frequency is double that of the voltage's.

</p><p><br>

</p>

<div class="mw-heading mw-heading3"><h3 id="Examples_of_alternating_current">Examples of alternating current</h3></div>

<p>To illustrate these concepts, consider a 230 V AC <a href="Mains_power_systems" class="mw-redirect" title="Mains power systems">mains</a> supply used in <a href="Mains_power_systems" class="mw-redirect" title="Mains power systems">many countries</a> around the world. It is so called because its <a href="Root_mean_square" title="Root mean square">root mean square</a> value is 230 V. This means that the time-averaged power delivered <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\text{average}}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{\text{average}}}</annotation>

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</math></span><img src="./4e36ebffc9c21b32af4d29e954ad6c146290ea14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.163ex; height:2.843ex;" alt="{\displaystyle P_{\text{average}}}" loading="lazy"></span> is equivalent to the power delivered by a DC voltage of 230 V. To determine the peak voltage (amplitude), we can rearrange the above equation to:

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<annotation encoding="application/x-tex">{\displaystyle V_{\text{peak}}={\sqrt {2}}\ V_{\text{rms}}}</annotation>

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</math></span><img src="./27beea29b4e53ae71e956ef7b0cc8161bd2fd9cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.948ex; height:3.343ex;" alt="{\displaystyle V_{\text{peak}}={\sqrt {2}}\ V_{\text{rms}}}" loading="lazy"></span></dd>

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<annotation encoding="application/x-tex">{\displaystyle P_{\text{peak}}={\frac {(V_{\text{rms}})^{2}}{R}}{\frac {(V_{\text{peak}})^{2}}{(V_{\text{rms}})^{2}}}={\text{P}}_{\text{average}}{\sqrt {2}}^{2}={\text{2}}P_{\text{average}}.}</annotation>

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</math></span><img src="./6d9eb63ac85af7b13059dba1d83d43d778ecffff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:51.302ex; height:6.676ex;" alt="{\displaystyle P_{\text{peak}}={\frac {(V_{\text{rms}})^{2}}{R}}{\frac {(V_{\text{peak}})^{2}}{(V_{\text{rms}})^{2}}}={\text{P}}_{\text{average}}{\sqrt {2}}^{2}={\text{2}}P_{\text{average}}.}" loading="lazy"></span></dd></dl>

<p>For 230 V AC, the peak voltage <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{\text{peak}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{\text{peak}}}</annotation>

</semantics>

</math></span><img src="./57804b7a3ccbc7cf3a67e709854cdd7919aab732.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.921ex; height:2.843ex;" alt="{\displaystyle V_{\text{peak}}}" loading="lazy"></span> is therefore <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 230{\text{ V}}\times {\sqrt {2}}}">

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 230{\text{ V}}\times {\sqrt {2}}}</annotation>

</semantics>

</math></span><img src="./a62b346a4d3e2423775bceec63970163ac5e7f59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.75ex; height:3.009ex;" alt="{\displaystyle 230{\text{ V}}\times {\sqrt {2}}}" loading="lazy"></span>, which is about 325 V, and the peak power <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{\text{peak}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P_{\text{peak}}}</annotation>

</semantics>

</math></span><img src="./13fd01a515155de886a12fef814897c6766a7a58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.058ex; height:2.843ex;" alt="{\displaystyle P_{\text{peak}}}" loading="lazy"></span> is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 230\times R\times W\times 2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 230\times R\times W\times 2}</annotation>

</semantics>

</math></span><img src="./d916ce83cdc4853d4b343368e019dc5754321a69.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.37ex; height:2.176ex;" alt="{\displaystyle 230\times R\times W\times 2}" loading="lazy"></span>, that is 460 RW. During the course of one cycle (two cycle as the power) the voltage rises from zero to 325 V, the power from zero to 460 RW, and both falls through zero. Next, the voltage descends to reverse direction, −325 V, but the power ascends again to 460 RW, and both returns to zero.

</p>

<div class="mw-heading mw-heading2"><h2 id="Information_transmission">Information transmission</h2></div>

<p>Alternating current is used to transmit <a href="Information" title="Information">information</a>, as in the cases of <a href="Telephone" title="Telephone">telephone</a> and <a href="Cable_television" title="Cable television">cable television</a>. Information signals are carried over a wide range of AC frequencies. <a href="Plain_old_telephone_service" title="Plain old telephone service">POTS</a> telephone signals have a frequency of about 3 kHz, close to the <a href="Baseband" title="Baseband">baseband</a> audio frequency. Cable television and other cable-transmitted information currents may alternate at frequencies of tens to thousands of megahertz. These frequencies are similar to the electromagnetic wave frequencies often used to transmit the same types of information <a href="Wireless" title="Wireless">over the air</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<p>The first <a href="Alternator" title="Alternator">alternator</a> to produce alternating current was an electric generator based on <a href="Michael_Faraday" title="Michael Faraday">Michael Faraday</a>'s principles constructed by the French instrument maker <a href="Hippolyte_Pixii" title="Hippolyte Pixii">Hippolyte Pixii</a> in 1832.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Pixii later added a <a href="Commutator_(electric)" title="Commutator (electric)">commutator</a> to his device to produce the (then) more commonly used direct current. The earliest recorded practical application of alternating current is by <a href="Guillaume_Duchenne" class="mw-redirect" title="Guillaume Duchenne">Guillaume Duchenne</a>, inventor and developer of <a href="Electrotherapy" title="Electrotherapy">electrotherapy</a>. In 1855, he announced that AC was superior to <a href="Direct_current" title="Direct current">direct current</a> for electrotherapeutic triggering of muscle contractions.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Alternating current technology was developed further by the Hungarian <a href="Ganz_Works" title="Ganz Works">Ganz Works</a> company in the 1870s, and, in the 1880s, by <a href="Sebastian_Ziani_de_Ferranti" title="Sebastian Ziani de Ferranti">Sebastian Ziani de Ferranti</a>, <a href="Lucien_Gaulard" title="Lucien Gaulard">Lucien Gaulard</a>, and <a href="Galileo_Ferraris" title="Galileo Ferraris">Galileo Ferraris</a>.

</p><p>In 1876, Russian engineer <a href="Pavel_Yablochkov" title="Pavel Yablochkov">Pavel Yablochkov</a> invented a lighting system where sets of induction coils were installed along a high-voltage AC line. Instead of changing voltage, the primary windings transferred power to the secondary windings which were connected to one or several <a href="Electric_candle" class="mw-redirect" title="Electric candle">electric candles</a> (arc lamps) of his own design,<sup id="cite_ref-maglab_5-0" class="reference"><a href="#cite_note-maglab-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> used to keep the failure of one lamp from disabling the entire circuit.<sup id="cite_ref-maglab_5-1" class="reference"><a href="#cite_note-maglab-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> In 1878, the <a href="Ganz_Works" title="Ganz Works">Ganz factory</a>, Budapest, Hungary, began manufacturing equipment for electric lighting and, by 1883, had installed over fifty systems in <a href="Austria-Hungary" title="Austria-Hungary">Austria-Hungary</a>. Their AC systems used arc and incandescent lamps, generators, and other equipment.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Transformers">Transformers</h3></div>

<p>The development of the alternating current <a href="Transformer" title="Transformer">transformer</a> to change voltage from low to high level and back, allowed generation and consumption at low voltages and transmission, over great distances, at high voltage, with savings in the cost of conductors and energy losses. A bipolar open-core <a href="Transformer" title="Transformer">power transformer</a> developed by <a href="Lucien_Gaulard" title="Lucien Gaulard">Lucien Gaulard</a> and <a href="John_Dixon_Gibbs" title="John Dixon Gibbs">John Dixon Gibbs</a> was demonstrated in London in 1881, and attracted the interest of <a href="Westinghouse_Electric_(1886)" class="mw-redirect" title="Westinghouse Electric (1886)">Westinghouse</a>. They exhibited an AC system powering arc and incandescent lights was installed along five railway stations for the Metropolitan Railway in <a href="London" title="London">London</a> and a single-phase multiple-user AC distribution system <a href="Turin" title="Turin">Turin</a> in 1884.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> These early induction coils with open magnetic circuits were inefficient at transferring power to <a href="Electrical_load" title="Electrical load">loads</a>. Until about 1880, the paradigm for AC power transmission from a high voltage supply to a low voltage load was a series circuit. Open-core transformers with a ratio near 1:1 were connected with their primaries in series to allow use of a high voltage for transmission while presenting a low voltage to the lamps. The inherent flaw in this method was that turning off a single lamp (or other electric device) affected the voltage supplied to all others on the same circuit. Many adjustable transformer designs were introduced to compensate for this problematic characteristic of the series circuit, including those employing methods of adjusting the core or bypassing the magnetic flux around part of a coil.<sup id="cite_ref-FJU1889_9-0" class="reference"><a href="#cite_note-FJU1889-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The direct current systems did not have these drawbacks, giving it significant advantages over early AC systems.

</p><p>In the UK, <a href="Sebastian_Ziani_de_Ferranti" title="Sebastian Ziani de Ferranti">Sebastian de Ferranti</a>, who had been developing AC generators and transformers in London since 1882, redesigned the AC system at the <a href="Grosvenor_Gallery#Generating_station" title="Grosvenor Gallery">Grosvenor Gallery power station</a> in 1886 for the London Electric Supply Corporation (LESCo) including alternators of his own design and open core transformer designs with serial connections for utilization loads - similar to Gaulard and Gibbs.<sup id="cite_ref-FOOTNOTEHughes199398_10-0" class="reference"><a href="#cite_note-FOOTNOTEHughes199398-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> In 1890, he designed <a href="Deptford_Power_Station" title="Deptford Power Station">their power station at Deptford</a><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> and converted the Grosvenor Gallery station across the Thames into an <a href="Electrical_substation" class="mw-redirect" title="Electrical substation">electrical substation</a>, showing the way to integrate older plants into a universal AC supply system.<sup id="cite_ref-FOOTNOTEHughes1993208_12-0" class="reference"><a href="#cite_note-FOOTNOTEHughes1993208-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>

</p>

<p>In the autumn of 1884, <a href="K%C3%A1roly_Zipernowsky" title="Károly Zipernowsky">Károly Zipernowsky</a>, <a href="Ott%C3%B3_Bl%C3%A1thy" title="Ottó Bláthy">Ottó Bláthy</a> and <a href="Miksa_D%C3%A9ri" title="Miksa Déri">Miksa Déri</a> (ZBD), three engineers associated with the <a href="Ganz_Works" title="Ganz Works">Ganz Works</a> of Budapest, determined that open-core devices were impractical, as they were incapable of reliably regulating voltage.<sup id="cite_ref-FOOTNOTEHughes199395_13-0" class="reference"><a href="#cite_note-FOOTNOTEHughes199395-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Bláthy had suggested the use of closed cores, Zipernowsky had suggested the use of <a href="Shunt_(electrical)" title="Shunt (electrical)">parallel shunt connections</a>, and Déri had performed the experiments;<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> In their joint 1885 patent applications for novel transformers (later called ZBD transformers), they described two designs with closed magnetic circuits where copper windings were either wound around a ring core of iron wires or else surrounded by a core of iron wires.<sup id="cite_ref-FJU1889_9-1" class="reference"><a href="#cite_note-FJU1889-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> In both designs, the magnetic flux linking the primary and secondary windings traveled almost entirely within the confines of the iron core, with no intentional path through air (see <a href="Transformer#Toroidal_cores" title="Transformer">toroidal cores</a>). The new transformers were 3.4 times more efficient than the open-core bipolar devices of Gaulard and Gibbs.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The Ganz factory in 1884 shipped the world's first five high-efficiency AC transformers.<sup id="cite_ref-Halacsy_(1961)_16-0" class="reference"><a href="#cite_note-Halacsy_(1961)-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> This first unit had been manufactured to the following specifications: 1,400 W, 40 Hz, 120:72 V, 11.6:19.4 A, ratio 1.67:1, one-phase, shell form.<sup id="cite_ref-Halacsy_(1961)_16-1" class="reference"><a href="#cite_note-Halacsy_(1961)-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup>

</p><p>The ZBD patents included two other major interrelated innovations: one concerning the use of parallel connected, instead of series connected, utilization loads, the other concerning the ability to have high turns ratio transformers such that the supply network voltage could be much higher (initially 140 to 2000 V) than the voltage of utilization loads (100 V initially preferred).<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> When employed in parallel connected electric distribution systems, closed-core transformers finally made it technically and economically feasible to provide electric power for lighting in homes, businesses and public spaces.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup>

</p><p>The other essential milestone was the introduction of 'voltage source, voltage intensive' (VSVI) systems'<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> by the invention of constant voltage generators in 1885.<sup id="cite_ref-FOOTNOTEHughes199396_22-0" class="reference"><a href="#cite_note-FOOTNOTEHughes199396-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> In early 1885, the three engineers also eliminated the problem of <a href="Eddy_current" title="Eddy current">eddy current</a> losses with the invention of the lamination of electromagnetic cores.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Ottó Bláthy also invented the first AC <a href="Electricity_meter" title="Electricity meter">electricity meter</a>.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Adoption">Adoption</h3></div>

<p>The AC power system was developed and adopted rapidly after 1886. In March of that year, Westinghouse engineer <a href="William_Stanley%2C_Jr." class="mw-redirect" title="William Stanley, Jr.">William Stanley</a>, designing a system based on the Gaulard and Gibbs transformer,<sup id="cite_ref-28" class="reference"><a href="#cite_note-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> demonstrated a lighting system in <a href="Great_Barrington%2C_Massachusetts" title="Great Barrington, Massachusetts">Great Barrington</a>: A <a href="Siemens" title="Siemens">Siemens</a> generator's voltage of 500 volts was converted into 3000 volts, and then the voltage was stepped down to 500 volts by six Westinghouse transformers. With this setup, the Westinghouse company successfully powered thirty 100-volt incandescent bulbs in twenty shops along the main street of Great Barrington.<sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> By the fall of that year Ganz engineers installed a ZBD transformer power system with AC generators in <a href="Rome" title="Rome">Rome</a>.<sup id="cite_ref-IEC_Techline_31-0" class="reference"><a href="#cite_note-IEC_Techline-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup>

</p>

<p>Based on Stanley's success, the new <a href="Westinghouse_Electric_Corporation" title="Westinghouse Electric Corporation">Westinghouse Electric</a><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> went on to develop alternating current (AC) electric infrastructure throughout the United States. The spread of Westinghouse and other AC systems triggered a push back in late 1887 by <a href="Thomas_Edison" title="Thomas Edison">Thomas Edison</a> (a proponent of direct current), who attempted to discredit alternating current as too dangerous in a public campaign called the "<a href="War_of_the_currents" title="War of the currents">war of the currents</a>".

</p><p>In 1888, alternating current systems gained further viability with the introduction of a functional <a href="AC_motor" title="AC motor">AC motor</a>, something these systems had lacked up till then. The design, an <a href="Induction_motor" title="Induction motor">induction motor</a>, was independently invented by <a href="Galileo_Ferraris" title="Galileo Ferraris">Galileo Ferraris</a> and <a href="Nikola_Tesla" title="Nikola Tesla">Nikola Tesla</a> (with Tesla's design being licensed by Westinghouse in the US). This design was independently further developed into the modern practical <a href="Three-phase" class="mw-redirect" title="Three-phase">three-phase</a> form by <a href="Mikhail_Dolivo-Dobrovolsky" title="Mikhail Dolivo-Dobrovolsky">Mikhail Dolivo-Dobrovolsky</a> and <a href="Charles_Eugene_Lancelot_Brown" title="Charles Eugene Lancelot Brown">Charles Eugene Lancelot Brown</a> in Germany on one side,<sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> and <a href="Jonas_Wenstr%C3%B6m" title="Jonas Wenström">Jonas Wenström</a> in Sweden on the other, though Brown favored the two-phase system.

</p><p>The <a href="Ames_Hydroelectric_Generating_Plant" title="Ames Hydroelectric Generating Plant">Ames Hydroelectric Generating Plant</a>, constructed in 1890, was among the first hydroelectric alternating current power plants. A long-distance transmission of single-phase electricity from a hydroelectric generating plant in Oregon at Willamette Falls sent power fourteen miles downriver to downtown Portland for street lighting in 1890.<sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> In 1891, another transmission system was installed in Telluride Colorado.<sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> The first <a href="Three-phase_electric_power" title="Three-phase electric power">three-phase system</a> was established in 1891 in <a href="Frankfurt" title="Frankfurt">Frankfurt</a>, Germany. The <a href="Tivoli%2C_Lazio" title="Tivoli, Lazio">Tivoli</a>–<a href="Rome" title="Rome">Rome</a> transmission was completed in 1892.<sup id="cite_ref-Holjevac_36-0" class="reference"><a href="#cite_note-Holjevac-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> The San Antonio Canyon Generator was the third commercial single-phase hydroelectric AC power plant in the United States to provide long-distance electricity. It was completed on December 31, 1892, by <a href="Almarian_Decker" title="Almarian Decker">Almarian William Decker</a> to provide power to the city of <a href="Pomona%2C_California" title="Pomona, California">Pomona, California</a>, which was 14 miles away. Meanwhile, the possibility of transferring electrical power from a waterfall at a distance was explored at the <a href="Gr%C3%A4ngesberg" title="Grängesberg">Grängesberg</a> mine in Sweden. A <span class="nowrap">45 <a href="Metre" title="Metre">m</a></span> fall at Hällsjön, Smedjebackens kommun, where a small iron work had been located, was selected. In 1893, a three-phase <span class="nowrap">9.5 <a href="Kilovolt" class="mw-redirect" title="Kilovolt">kv</a></span> system was used to transfer 400 <a href="Horsepower" title="Horsepower">horsepower</a> a distance of <span class="nowrap">15 <a href="Kilometre" title="Kilometre">km</a></span>, becoming the first commercial application.<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> In 1893, Westinghouse built an alternating current system for the <a href="Chicago_World_Exposition" class="mw-redirect" title="Chicago World Exposition">Chicago World Exposition</a>.<sup id="cite_ref-Holjevac_36-1" class="reference"><a href="#cite_note-Holjevac-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> In 1893, Decker designed the first American commercial <a href="Three-phase" class="mw-redirect" title="Three-phase">three-phase</a> power plant using alternating current—the hydroelectric <a href="Mill_Creek_No._1_Hydroelectric_Plant" class="mw-redirect" title="Mill Creek No. 1 Hydroelectric Plant">Mill Creek No. 1 Hydroelectric Plant</a> near <a href="Redlands%2C_California" title="Redlands, California">Redlands, California</a>. Decker's design incorporated 10 kV three-phase transmission and established the standards for the complete system of generation, transmission and motors used in USA today. The original Niagara Falls <a href="Adams_Power_Plant" class="mw-redirect" title="Adams Power Plant">Adams Power Plant</a> with three two-phase generators was put into operation in August 1895, but was connected to the remote transmission system only in 1896. The <a href="Jaruga_Hydroelectric_Power_Plant" title="Jaruga Hydroelectric Power Plant">Jaruga Hydroelectric Power Plant</a> in Croatia was set in operation two days later, on 28 August 1895. Its <a href="Electric_generator" title="Electric generator">generator</a> (42 Hz, 240 kW) was made and installed by the Hungarian company <a href="Ganz" class="mw-redirect" title="Ganz">Ganz</a>, while the transmission line from the power plant to the City of <a href="%C5%A0ibenik" title="Šibenik">Šibenik</a> was 11.5 kilometers (7.1 mi) long, and the municipal distribution grid 3000 V/110 V included six transforming stations.<sup id="cite_ref-Holjevac_36-2" class="reference"><a href="#cite_note-Holjevac-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup>

</p><p>Alternating current circuit theory developed rapidly in the latter part of the 19th and early 20th century. Notable contributors to the theoretical basis of alternating current calculations include <a href="Charles_Steinmetz" class="mw-redirect" title="Charles Steinmetz">Charles Steinmetz</a>, <a href="Oliver_Heaviside" title="Oliver Heaviside">Oliver Heaviside</a>, and many others.<sup id="cite_ref-38" class="reference"><a href="#cite_note-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> Calculations in unbalanced three-phase systems were simplified by the <a href="Symmetrical_components" title="Symmetrical components">symmetrical components</a> methods discussed by <a href="Charles_LeGeyt_Fortescue" title="Charles LeGeyt Fortescue">Charles LeGeyt Fortescue</a> in 1918.

</p>

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<ul><li><a href="AC_power" title="AC power">AC power</a></li>

<li><a href="Electrical_wiring" title="Electrical wiring">Electrical wiring</a></li>

<li><a href="Industrial_and_multiphase_power_plugs_and_sockets" title="Industrial and multiphase power plugs and sockets">Heavy-duty power plugs</a></li>

<li><a href="Hertz" title="Hertz">Hertz</a></li>

<li><a href="Leading_and_lagging_current" title="Leading and lagging current">Leading and lagging current</a></li>

<li><a href="Mains_electricity_by_country" title="Mains electricity by country">Mains electricity by country</a></li>

<li><a href="AC_power_plugs_and_sockets" title="AC power plugs and sockets">AC power plugs and sockets</a></li>

<li><a href="Utility_frequency" title="Utility frequency">Utility frequency</a></li>

<li><a href="War_of_the_currents" title="War of the currents">War of the currents</a></li>

<li><a href="AC/DC_receiver_design" title="AC/DC receiver design">AC/DC receiver design</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-FOOTNOTEHughes199398-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHughes199398_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHughes1993">Hughes (1993)</a>, p. 98.</span>

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<li id="cite_note-FOOTNOTEHughes1993208-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHughes1993208_12-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHughes1993">Hughes (1993)</a>, p. 208.</span>

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<li id="cite_note-FOOTNOTEHughes199395-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEHughes199395_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFHughes1993">Hughes (1993)</a>, p. 95.</span>

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<li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text"><cite id="CITEREFBrussoAllerhand2021" class="citation journal cs1">Brusso, Barry; Allerhand, Adam (January 2021). <a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FMIAS.2020.3028630">"A Contrarian History of Early Electric Power Distribution [History]"</a>. <i>IEEE Industry Applications Magazine</i>. <b>27</b>. IEEE.org: 13. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FMIAS.2020.3028630">10.1109/MIAS.2020.3028630</a></span>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:230605234">230605234</a>.</cite></span>

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<li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text"><cite id="CITEREFClark_W._Gellings2020" class="citation book cs1">Clark W. Gellings (2020). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0xkOEAAAQBAJ&dq=siemens+generator+%22great+barrington%22&pg=PT62"><i>The Smart Grid Enabling Energy Efficiency and Demand Response</i></a>. River Publishers. p. 62. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781000355314</bdi>.</cite></span>

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<li id="cite_note-IEC_Techline-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-IEC_Techline_31-0">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070930171011/http://www.iec.ch/cgi-bin/tl_to_htm.pl?section=technology&item=144">"Ottó Bláthy, Miksa Déri, Károly Zipernowsky"</a>. IEC Techline. Archived from <a rel="nofollow" class="external text" href="http://www.iec.ch/cgi-bin/tl_to_htm.pl?section=technology&item=144">the original</a> on September 30, 2007<span class="reference-accessdate">. Retrieved <span class="nowrap">Apr 16,</span> 2010</span>.</cite></span>

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<li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="http://tthsdelco.org/wp-content/uploads/2014/02/History%20of%20Tinicum%20Twp.pdf"><i>History of Tinicum Township (PA) 1643–1993</i></a> <span class="cs1-format">(PDF)</span>. Tinicum Township Historical Society. 1993. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20150423202458/http://tthsdelco.org/wp-content/uploads/2014/02/History%20of%20Tinicum%20Twp.pdf">Archived</a> <span class="cs1-format">(PDF)</span> from the original on April 23, 2015.</cite></span>

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<li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><cite id="CITEREFHeertjePerlman1990" class="citation book cs1">Heertje, Arnold; Perlman, Mark (1990). <i>Evolving Technology and Market Structure: Studies in Schumpeterian Economics</i>. University of Michigan Press. p. 138. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780472101924</bdi>.</cite></span>

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<li id="cite_note-34"><span class="mw-cite-backlink"><b><a href="#cite_ref-34">^</a></b></span> <span class="reference-text"><cite class="citation journal cs1">"Electric Transmission of Power". <i>General Electric Review</i>. <b>XVIII</b>. 1915.</cite></span>

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<li id="cite_note-35"><span class="mw-cite-backlink"><b><a href="#cite_ref-35">^</a></b></span> <span class="reference-text"><cite class="citation journal cs1">"Electric Transmission of Power". <i>General Electric</i>. <b>XVIII</b>. 1915.</cite></span>

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<li id="cite_note-Holjevac-36"><span class="mw-cite-backlink">^ <a href="#cite_ref-Holjevac_36-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Holjevac_36-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Holjevac_36-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFHoljevacKuzle2019" class="citation journal cs1 cs1-prop-foreign-lang-source">Holjevac, Ninoslav; Kuzle, Igor (2019). <a rel="nofollow" class="external text" href="https://hrcak.srce.hr/238713">"Prvi cjeloviti višefazni elektroenergetski sustav na svijetu – Krka Šibenik"</a>. <i>Godišnjak Akademije tehničkih znanosti Hrvatske</i> (in Croatian). <b>2019</b> (1): <span class="nowrap">162–</span>174. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2975-657X">2975-657X</a>.</cite></span>

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<li id="cite_note-37"><span class="mw-cite-backlink"><b><a href="#cite_ref-37">^</a></b></span> <span class="reference-text"><cite id="CITEREFHjulström1940" class="citation book cs1">Hjulström, Filip (1940). <a rel="nofollow" class="external text" href="https://www.antikvariat.net/sv/rod151504-elektrifieringens-utveckling-i-sverige-en-ekonomisk-geografisk-oversikt-hjulstrom-filip"><i>Elektrifieringens utveckling i Sverige, en ekonomisk-geografisk översikt</i></a>. [Excerpt taken from YMER 1941, häfte 2.Utgiven av Sällskapet för antropologi och geografi: Meddelande från Upsala univeristets geografiska institution, N:o 29, published by Esselte ab, Stockholm 1941 no. 135205]</cite></span>

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<li id="cite_note-38"><span class="mw-cite-backlink"><b><a href="#cite_ref-38">^</a></b></span> <span class="reference-text"><cite id="CITEREFGrattan-Guinness2003" class="citation book cs1">Grattan-Guinness, I. (September 19, 2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=f5FqsDPVQ2MC&q=theoretical++alternating+current++Oliver+Heaviside&pg=PA1229"><i>Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences</i></a>. JHU Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8018-7397-3</bdi> – via Google Books.</cite></span>

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<li id="cite_note-39"><span class="mw-cite-backlink"><b><a href="#cite_ref-39">^</a></b></span> <span class="reference-text"><cite id="CITEREFSuzuki2009" class="citation book cs1">Suzuki, Jeff (August 27, 2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lew5IC5piCwC&q=theoretical++alternating+current++Charles+Steinmetz&pg=PA329"><i>Mathematics in Historical Context</i></a>. MAA. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-88385-570-6</bdi> – via Google Books.</cite></span>

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</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

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<ul><li>Willam A. Meyers, <i>History and Reflections on the Way Things Were: Mill Creek Power Plant – Making History with AC</i>, IEEE Power Engineering Review, February 1997, pp. 22–24</li></ul>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Alternating_current" class="extiw external" title="commons:Category:Alternating current">Alternating current</a></span>.</div></div>

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<ul><li>"<i>AC/DC: <a rel="nofollow" class="external text" href="https://archive.today/20121204180831/http://www.pbs.org/wgbh/amex/edison/sfeature/acdc.html">What's the Difference</a>?</i>". Edison's Miracle of Light, <a rel="nofollow" class="external text" href="https://web.archive.org/web/19991128155408/http://www.pbs.org/wgbh/amex/index.html">American Experience</a>. (<a href="Public_Broadcasting_Service" class="mw-redirect" title="Public Broadcasting Service">PBS</a>)</li>

<li>"<i>AC/DC: <a rel="nofollow" class="external text" href="https://www.pbs.org/wgbh/amex/edison/sfeature/acdc_insideacgenerator.html">Inside the AC Generator</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141228182024/http://www.pbs.org/wgbh/amex/edison/sfeature/acdc_insideacgenerator.html">Archived</a> 2014-12-28 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></i>". Edison's Miracle of Light, American Experience. (PBS)</li>

<li><a rel="nofollow" class="external text" href="http://www.technology.niagarac.on.ca/people/mcsele/Rankine.html">Professor Mark Csele's tour of the 25 Hz Rankine generating station</a></li>

<li>Blalock, Thomas J., "<i><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070607042254/http://www.ieee.org/organizations/pes/public/2003/sep/peshistory.html">The Frequency Changer Era: Interconnecting Systems of Varying Cycles</a></i>". The history of various frequencies and interconversion schemes in the US at the beginning of the 20th century</li>

<li><a rel="nofollow" class="external text" href="http://edisontechcenter.org/AC-PowerHistory.html">AC Power History and Timeline</a></li></ul>

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