Modulation

From Wikipedia, the free encyclopedia

In electronics and telecommunications, modulation is the process of varying one or more properties of a periodic waveform, called the carrier signal, with a modulating signal that typically contains information to be transmitted. Most radio systems in the 20th century used frequency modulation (FM) or amplitude modulation (AM) to make the carrier carry the radio broadcast.

In general telecommunications, modulation is a process of conveying message signal, for example, a digital bit stream or an analog audio signal, inside another signal that can be physically transmitted. Modulation of a sine waveform transforms a narrow frequency range baseband message signal into a moderate to high frequency range passband signal, one that can pass through a filter.

A modulator is a device that performs modulation. A demodulator (sometimes detector or demod) is a device that performs demodulation, the inverse of modulation. A modem (from modulator–demodulator) can perform both operations.

The aim of analog modulation is to transfer an analog baseband (or lowpass) signal, for example an audio signal or TV signal, over an analog bandpass channel at a different frequency, for example over a limited radio frequency band or a cable TV network channel.

The aim of digital modulation is to transfer a digital bit stream over an analog communication channel, for example over the public switched telephone network (where a bandpass filter limits the frequency range to 300–3400 Hz) or over a limited radio frequency band.

Analog and digital modulation facilitate frequency division multiplexing (FDM), where several low pass information signals are transferred simultaneously over the same shared physical medium, using separate passband channels (several different carrier frequencies).

The aim of digital baseband modulation methods, also known as line coding, is to transfer a digital bit stream over a baseband channel, typically a non-filtered copper wire such as a serial bus or a wired local area network.

The aim of pulse modulation methods is to transfer a narrowband analog signal, for example, a phone call over a wideband baseband channel or, in some of the schemes, as a bit stream over another digital transmission system.

In music synthesizers, modulation may be used to synthesize waveforms with an extensive overtone spectrum using a small number of oscillators. In this case, the carrier frequency is typically in the same order or much lower than the modulating waveform (see frequency modulation synthesis or ring modulation synthesis).

Analog modulation methods

A low-frequency message signal (top) may be carried by an AM or FM radio wave.

 

Waterfall plot of a 146.52 MHZ radio carrier, with amplitude modulation by a 1,000 hz sinusoid. Two strong sidebands at + and - 1Khz from the carrier frequency are shown.

 

A carrier, frequency modulated by a 1,000 hz sinusoid. The modulation index has been adjusted to around 2.4, so the carrier frequency has small amplitude. Several strong sidebands are apparent; in principle an infinite number are produced in FM but the higher-order sidebands are of negligible magnitude.

In analog modulation, the modulation is applied continuously in response to the analog information signal. Common analog modulation techniques are include:

• Amplitude modulation (AM) (here the amplitude of the carrier signal is varied in accordance with the instantaneous amplitude of the modulating signal)
  • Double-sideband modulation (DSB)
    • Double-sideband modulation with carrier (DSB-WC) (used on the AM radio broadcasting band)
    • Double-sideband suppressed-carrier transmission (DSB-SC)
    • Double-sideband reduced carrier transmission (DSB-RC)
  • Single-sideband modulation (SSB, or SSB-AM)
    • Single-sideband modulation with carrier (SSB-WC)
    • Single-sideband modulation suppressed carrier modulation (SSB-SC)
  • Vestigial sideband modulation (VSB, or VSB-AM)
  • Quadrature amplitude modulation (QAM)
• Angle modulation, which is approximately constant envelope
  • Frequency modulation (FM) (here the frequency of the carrier signal is varied in accordance with the instantaneous amplitude of the modulating signal)
  • Phase modulation (PM) (here the phase shift of the carrier signal is varied in accordance with the instantaneous amplitude of the modulating signal)
  • Transpositional Modulation (TM), in which the waveform inflection is modified resulting in a signal where each quarter cycle is transposed in the modulation process. TM is a pseudo-analog modulation (AM). Where an AM carrier also carries a phase variable phase f(ǿ). TM is f(AM,ǿ)

Digital modulation methods

In digital modulation, an analog carrier signal is modulated by a discrete signal. Digital modulation methods can be considered as digital-to-analog conversion and the corresponding demodulation or detection as analog-to-digital conversion. The changes in the carrier signal are chosen from a finite number of M alternative symbols (the modulation alphabet).

Schematic of 4 baud (8 bit/s) data link containing arbitrarily chosen values.

A simple example: A telephone line is designed for transferring audible sounds, for example, tones, and not digital bits (zeros and ones). Computers may, however, communicate over a telephone line by means of modems, which are representing the digital bits by tones, called symbols. If there are four alternative symbols (corresponding to a musical instrument that can generate four different tones, one at a time), the first symbol may represent the bit sequence 00, the second 01, the third 10 and the fourth 11. If the modem plays a melody consisting of 1000 tones per second, the symbol rate is 1000 symbols/second, or baud. Since each tone (i.e., symbol) represents a message consisting of two digital bits in this example, the bit rate is twice the symbol rate, i.e. 2000 bits per second. This is similar to the technique used by dial-up modems as opposed to DSL modems.

According to one definition of digital signal, the modulated signal is a digital signal. According to another definition, the modulation is a form of digital-to-analog conversion. Most textbooks would consider digital modulation schemes as a form of digital transmission, synonymous to data transmission; very few would consider it as analog transmission.

Fundamental digital modulation methods

The most fundamental digital modulation techniques are based on keying:

• PSK (phase-shift keying): a finite number of phases are used.
• FSK (frequency-shift keying): a finite number of frequencies are used.
• ASK (amplitude-shift keying): a finite number of amplitudes are used.
• QAM (quadrature amplitude modulation): a finite number of at least two phases and at least two amplitudes are used.

In QAM, an in-phase signal (or I, with one example being a cosine waveform) and a quadrature phase signal (or Q, with an example being a sine wave) are amplitude modulated with a finite number of amplitudes and then summed. It can be seen as a two-channel system, each channel using ASK. The resulting signal is equivalent to a combination of PSK and ASK.

In all of the above methods, each of these phases, frequencies or amplitudes are assigned a unique pattern of binary bits. Usually, each phase, frequency or amplitude encodes an equal number of bits. This number of bits comprises the symbol that is represented by the particular phase, frequency or amplitude.

If the alphabet consists of ${\displaystyle M=2^{N}}$ alternative symbols, each symbol represents a message consisting of N bits. If the symbol rate (also known as the baud rate) is ${\displaystyle f_{S}}$ symbols/second (or baud), the data rate is ${\displaystyle Nf_{S}}$ bit/second.

For example, with an alphabet consisting of 16 alternative symbols, each symbol represents 4 bits. Thus, the data rate is four times the baud rate.

In the case of PSK, ASK or QAM, where the carrier frequency of the modulated signal is constant, the modulation alphabet is often conveniently represented on a constellation diagram, showing the amplitude of the I signal at the x-axis, and the amplitude of the Q signal at the y-axis, for each symbol.

Modulator and detector principles of operation

PSK and ASK, and sometimes also FSK, are often generated and detected using the principle of QAM. The I and Q signals can be combined into a complex-valued signal I+jQ (where j is the imaginary unit). The resulting so called equivalent lowpass signal or equivalent baseband signal is a complex-valued representation of the real-valued modulated physical signal (the so-called passband signal or RF signal).

These are the general steps used by the modulator to transmit data:

1. Group the incoming data bits into codewords, one for each symbol that will be transmitted.
2. Map the codewords to attributes, for example, amplitudes of the I and Q signals (the equivalent low pass signal), or frequency or phase values.
3. Adapt pulse shaping or some other filtering to limit the bandwidth and form the spectrum of the equivalent low pass signal, typically using digital signal processing.
4. Perform digital to analog conversion (DAC) of the I and Q signals (since today all of the above is normally achieved using digital signal processing, DSP).
5. Generate a high-frequency sine carrier waveform, and perhaps also a cosine quadrature component. Carry out the modulation, for example by multiplying the sine and cosine waveform with the I and Q signals, resulting in the equivalent low pass signal being frequency shifted to the modulated passband signal or RF signal. Sometimes this is achieved using DSP technology, for example direct digital synthesis using a waveform table, instead of analog signal processing. In that case, the above DAC step should be done after this step.
6. Amplification and analog bandpass filtering to avoid harmonic distortion and periodic spectrum.

At the receiver side, the demodulator typically performs:

1. Bandpass filtering.
2. Automatic gain control, AGC (to compensate for attenuation, for example fading).
3. Frequency shifting of the RF signal to the equivalent baseband I and Q signals, or to an intermediate frequency (IF) signal, by multiplying the RF signal with a local oscillator sine wave and cosine wave frequency (see the superheterodyne receiver principle).
4. Sampling and analog-to-digital conversion (ADC) (sometimes before or instead of the above point, for example by means of undersampling).
5. Equalization filtering, for example, a matched filter, compensation for multipath propagation, time spreading, phase distortion and frequency selective fading, to avoid intersymbol interference and symbol distortion.
6. Detection of the amplitudes of the I and Q signals, or the frequency or phase of the IF signal.
7. Quantization of the amplitudes, frequencies or phases to the nearest allowed symbol values.
8. Mapping of the quantized amplitudes, frequencies or phases to codewords (bit groups).
9. Parallel-to-serial conversion of the codewords into a bit stream.
10. Pass the resultant bit stream on for further processing such as removal of any error-correcting codes.

As is common to all digital communication systems, the design of both the modulator and demodulator must be done simultaneously. Digital modulation schemes are possible because the transmitter-receiver pair has prior knowledge of how data is encoded and represented in the communications system. In all digital communication systems, both the modulator at the transmitter and the demodulator at the receiver are structured so that they perform inverse operations.

Non-coherent modulation methods do not require a receiver reference clock signal that is phase synchronized with the sender carrier signal. In this case, modulation symbols (rather than bits, characters, or data packets) are asynchronously transferred. The opposite is coherent modulation.

List of common digital modulation techniques

The most common digital modulation techniques are:

• Phase-shift keying (PSK)
  • Binary PSK (BPSK), using M=2 symbols
  • Quadrature PSK (QPSK), using M=4 symbols
  • 8PSK, using M=8 symbols
  • 16PSK, using M=16 symbols
  • Differential PSK (DPSK)
  • Differential QPSK (DQPSK)
  • Offset QPSK (OQPSK)
  • π/4–QPSK
• Frequency-shift keying (FSK)
  • Audio frequency-shift keying (AFSK)
  • Multi-frequency shift keying (M-ary FSK or MFSK)
  • Dual-tone multi-frequency (DTMF)
• Amplitude-shift keying (ASK)
• On-off keying (OOK), the most common ASK form
  • M-ary vestigial sideband modulation, for example 8VSB
• Quadrature amplitude modulation (QAM), a combination of PSK and ASK
  • Polar modulation like QAM a combination of PSK and ASK^{[citation needed]}
• Continuous phase modulation (CPM) methods
  • Minimum-shift keying (MSK)
  • Gaussian minimum-shift keying (GMSK)
  • Continuous-phase frequency-shift keying (CPFSK)
• Orthogonal frequency-division multiplexing (OFDM) modulation
  • Discrete multitone (DMT), including adaptive modulation and bit-loading
• Wavelet modulation
• Trellis coded modulation (TCM), also known as Trellis modulation
• Spread-spectrum techniques
  • Direct-sequence spread spectrum (DSSS)
  • Chirp spread spectrum (CSS) according to IEEE 802.15.4a CSS uses pseudo-stochastic coding
  • Frequency-hopping spread spectrum (FHSS) applies a special scheme for channel release

MSK and GMSK are particular cases of continuous phase modulation. Indeed, MSK is a particular case of the sub-family of CPM known as continuous-phase frequency shift keying (CPFSK) which is defined by a rectangular frequency pulse (i.e. a linearly increasing phase pulse) of one-symbol-time duration (total response signaling).

OFDM is based on the idea of frequency-division multiplexing (FDM), but the multiplexed streams are all parts of a single original stream. The bit stream is split into several parallel data streams, each transferred over its own sub-carrier using some conventional digital modulation scheme. The modulated sub-carriers are summed to form an OFDM signal. This dividing and recombining help with handling channel impairments. OFDM is considered as a modulation technique rather than a multiplex technique since it transfers one bit stream over one communication channel using one sequence of so-called OFDM symbols. OFDM can be extended to multi-user channel access method in the orthogonal frequency-division multiple access (OFDMA) and multi-carrier code division multiple access (MC-CDMA) schemes, allowing several users to share the same physical medium by giving different sub-carriers or spreading codes to different users.

Of the two kinds of RF power amplifier, switching amplifiers (Class D amplifiers) cost less and use less battery power than linear amplifiers of the same output power. However, they only work with relatively constant-amplitude-modulation signals such as angle modulation (FSK or PSK) and CDMA, but not with QAM and OFDM. Nevertheless, even though switching amplifiers are completely unsuitable for normal QAM constellations, often the QAM modulation principle are used to drive switching amplifiers with these FM and other waveforms, and sometimes QAM demodulators are used to receive the signals put out by these switching amplifiers.

Automatic digital modulation recognition (ADMR)

Automatic digital modulation recognition in intelligent communication systems is one of the most important issues in software defined radio and cognitive radio. According to incremental expanse of intelligent receivers, automatic modulation recognition becomes a challenging topic in telecommunication systems and computer engineering. Such systems have many civil and military applications. Moreover, blind recognition of modulation type is an important problem in commercial systems, especially in software defined radio. Usually in such systems, there are some extra information for system configuration, but considering blind approaches in intelligent receivers, we can reduce information overload and increase transmission performance. Obviously, with no knowledge of the transmitted data and many unknown parameters at the receiver, such as the signal power, carrier frequency and phase offsets, timing information, etc., blind identification of the modulation is made fairly difficult. This becomes even more challenging in real-world scenarios with multipath fading, frequency-selective and time-varying channels.

There are two main approaches to automatic modulation recognition. The first approach uses likelihood-based methods to assign an input signal to a proper class. Another recent approach is based on feature extraction.

Digital baseband modulation or line coding

The term digital baseband modulation (or digital baseband transmission) is synonymous to line codes. These are methods to transfer a digital bit stream over an analog baseband channel (a.k.a. lowpass channel) using a pulse train, i.e. a discrete number of signal levels, by directly modulating the voltage or current on a cable or serial bus. Common examples are unipolar, non-return-to-zero (NRZ), Manchester and alternate mark inversion (AMI) codings.

Pulse modulation methods

Pulse modulation schemes aim at transferring a narrowband analog signal over an analog baseband channel as a two-level signal by modulating a pulse wave. Some pulse modulation schemes also allow the narrowband analog signal to be transferred as a digital signal (i.e., as a quantized discrete-time signal) with a fixed bit rate, which can be transferred over an underlying digital transmission system, for example, some line code. These are not modulation schemes in the conventional sense since they are not channel coding schemes, but should be considered as source coding schemes, and in some cases analog-to-digital conversion techniques.

Analog-over-analog methods

• Pulse-amplitude modulation (PAM)
• Pulse-width modulation (PWM) and Pulse-depth modulation (PDM)
• Pulse-position modulation (PPM)

Analog-over-digital methods

• Pulse-code modulation (PCM)
  • Differential PCM (DPCM)
  • Adaptive DPCM (ADPCM)
• Delta modulation (DM or Δ-modulation)
• Delta-sigma modulation (∑Δ)
• Continuously variable slope delta modulation (CVSDM), also called Adaptive-delta modulation (ADM)
• Pulse-density modulation (PDM)

Miscellaneous modulation techniques

• The use of on-off keying to transmit Morse code at radio frequencies is known as continuous wave (CW) operation.
• Adaptive modulation
• Space modulation is a method whereby signals are modulated within airspace such as that used in instrument landing systems.

Alternating current

From Wikipedia, the free encyclopedia

Alternating current (green curve). The horizontal axis measures time; the vertical, current or voltage.

Alternating current (AC) is an electric current which periodically reverses direction, in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in which electric power is delivered to businesses and residences, and it is the form of electrical energy that consumers typically use when they plug kitchen appliances, televisions, fans and electric lamps into a wall socket. A common source of DC power is a battery cell in a flashlight. The abbreviations AC and DC are often used to mean simply alternating and direct, as when they modify current or voltage.

The usual waveform of alternating current (ac) in most electric power circuits is a sine wave. In certain applications, different waveforms are used, such as triangular or square waves. Audio and radio 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 modulation of an AC carrier signal. These currents typically alternate at higher frequencies than those used in power transmission.

Transmission, distribution, and domestic power supply

A schematic representation of long distance electric power transmission. C=consumers, D=step down transformer, G=generator, I=current in the wires, Pe=power reaching the end of the transmission line, Pt=power entering the transmission line, Pw=power lost in the transmission line, R=total resistance in the wires, V=voltage at the beginning of the transmission line, U=step up transformer.

Electrical energy is distributed as alternating current because AC voltage may be increased or decreased with a transformer. This allows the power to be transmitted through power lines efficiently at high voltage, which reduces the energy lost as heat due to resistance 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 (${\displaystyle P_{\rm {w}}}$) in the wire are a product of the square of the current (I) and the resistance (R) of the wire, described by the formula

${\displaystyle P_{\rm {w}}=I^{2}R\,.}$

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 will be four times less.

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

${\displaystyle P_{\rm {t}}=IV\,.}$

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, and transformed to 100 V – 240 V for domestic use.

High voltage transmission lines deliver power from electric generation plants over long distances using alternating current. These lines are located in eastern Utah.

High voltages have disadvantages, such as the increased insulation required, and generally increased difficulty in their safe handling. In a power plant, energy is generated at a convenient voltage for the design of a generator, 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 mains power systems found in the world. High-voltage direct-current (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 electric power transmission, as there was then no economically viable way to step down the voltage of DC for end user applications such as lighting incandescent bulbs.

Three-phase electrical generation is very common. The simplest way is to use three separate coils in the generator stator, 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° out of phase to each other. If coils are added opposite to these (60° spacing), they generate the same phases with reverse polarity and so can be simply wired together. In practice, higher "pole orders" 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 neutral point. 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 harmonics. Harmonics can cause neutral conductor current levels to exceed that of one or all phase conductors.

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 single phase 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. Three-wire single-phase 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 "two phase". 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 electric shock 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.

A third wire, 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 electrical impedance path to ground sufficient to carry any fault 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.

AC power supply frequencies

The frequency of the electrical system varies by country and sometimes within a country; most electric power is generated at either 50 or 60 Hertz. Some countries have a mixture of 50 Hz and 60 Hz supplies, notably electricity power transmission in Japan. A low frequency eases the design of electric motors, particularly for hoisting, crushing and rolling applications, and commutator-type traction motors for applications such as railways. However, low frequency also causes noticeable flicker in arc lamps and incandescent light bulbs. The use of lower frequencies also provided the advantage of lower impedance losses, which are proportional to frequency. 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 Austria, Germany, Norway, Sweden and Switzerland. 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 mainframe systems were often powered by 400 Hz or 415 Hz for benefits of ripple reduction while using smaller internal AC to DC conversion units. In any case, the input to the M-G set is the local customary voltage and frequency, variously 200 V (Japan), 208 V, 240 V (North America), 380 V, 400 V or 415 V (Europe), and variously 50 Hz or 60 Hz.

Effects at high frequencies

Play media

A Tesla coil producing high-frequency current that is harmless to humans, but lights a fluorescent lamp when brought near it

A direct current flows uniformly throughout the cross-section of a uniform wire. An alternating current of any frequency is forced away from the wire's center, toward its outer surface. This is because the acceleration of an electric charge in an alternating current produces waves of electromagnetic radiation that cancel the propagation of electricity toward the center of materials with high conductivity. This phenomenon is called skin effect. At very high frequencies the current no longer flows in the wire, but effectively flows on the surface of the wire, within a thickness of a few skin depths. 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 conductors. 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. Since the current tends to flow in the periphery of conductors, the effective cross-section of the conductor is reduced. This increases the effective AC resistance of the conductor, since resistance is inversely proportional to the cross-sectional area. The AC resistance often is many times higher than the DC resistance, causing a much higher energy loss due to ohmic heating (also called I²R loss).

Techniques for reducing AC resistance

For low to medium frequencies, conductors can be divided into stranded wires, each insulated from one another, and the relative positions of individual strands specially arranged within the conductor bundle. Wire constructed using this technique is called Litz wire. 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 high-Q inductors, reducing losses in flexible conductors carrying very high currents at lower frequencies, and in the windings of devices carrying higher radio frequency current (up to hundreds of kilohertz), such as switch-mode power supplies and radio frequency transformers.

Techniques for reducing radiation loss

As written above, an alternating current is made of electric charge under periodic acceleration, which causes radiation of electromagnetic waves. Energy that is radiated is lost. Depending on the frequency, different techniques are used to minimize the loss due to radiation.

Twisted pairs

At frequencies up to about 1 GHz, pairs of wires are twisted together in a cable, forming a twisted pair. This reduces losses from electromagnetic radiation and inductive coupling. A twisted pair must be used with a balanced signalling system, so that the two wires carry equal but opposite currents. Each wire in a twisted pair radiates a signal, but it is effectively cancelled by radiation from the other wire, resulting in almost no radiation loss.

Coaxial cables

Coaxial cables are commonly used at audio frequencies and above for convenience. A coaxial cable has a conductive wire inside a conductive tube, separated by a dielectric 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 microwave frequencies greater than 5 GHz, the losses (due mainly to the electrical resistance of the central conductor) become too large, making waveguides a more efficient medium for transmitting energy. Coaxial cables with an air rather than solid dielectric are preferred as they transmit power with lower loss.

Waveguides

Waveguides are similar to coaxial cables, as both consist of tubes, with the biggest difference being that the waveguide has 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 electric current, but rather by means of a guided electromagnetic field. Although surface currents 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 wavelength of the alternating current to be transmitted, so they are only feasible at microwave frequencies. In addition to this mechanical feasibility, electrical resistance of the non-ideal metals forming the walls of the waveguide cause dissipation of power (surface currents flowing on lossy conductors dissipate power). At higher frequencies, the power lost to this dissipation becomes unacceptably large.

Fiber optics

At frequencies greater than 200 GHz, waveguide dimensions become impractically small, and the ohmic losses in the waveguide walls become large. Instead, fiber optics, which are a form of dielectric waveguides, can be used. For such frequencies, the concepts of voltages and currents are no longer used.

Mathematics of AC voltages

A sinusoidal alternating voltage.
1 = Peak, also amplitude,
2 = Peak-to-peak,
3 = Effective value,
4 = Period

A sine wave, over one cycle (360°). The dashed line represents the root mean square (RMS) value at about 0.707

Alternating currents are accompanied (or caused) by alternating voltages. An AC voltage v can be described mathematically as a function of time by the following equation:

${\displaystyle v(t)=V_{\mathrm {peak} }\cdot \sin(\omega t)}$,

where

• ${\displaystyle \displaystyle V_{\rm {peak}}}$ is the peak voltage (unit: volt),
• ${\displaystyle \displaystyle \omega }$ is the angular frequency (unit: radians per second)
  • The angular frequency is related to the physical frequency, ${\displaystyle \displaystyle f}$ (unit = hertz), which represents the number of cycles per second, by the equation ${\displaystyle \displaystyle \omega =2\pi f}$.
• ${\displaystyle \displaystyle t}$ is the time (unit: second).

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 ${\displaystyle \sin(x)}$ is +1 and the minimum value is −1, an AC voltage swings between ${\displaystyle +V_{\rm {peak}}}$ and ${\displaystyle -V_{\rm {peak}}}$. The peak-to-peak voltage, usually written as ${\displaystyle V_{\rm {pp}}}$ or ${\displaystyle V_{\rm {P-P}}}$, is therefore ${\displaystyle V_{\rm {peak}}-(-V_{\rm {peak}})=2V_{\rm {peak}}}$.

Power

The relationship between voltage and the power delivered is

${\displaystyle p(t)={\frac {v^{2}(t)}{R}}}$ where ${\displaystyle R}$ represents a load resistance.

Rather than using instantaneous power, ${\displaystyle p(t)}$, 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 root mean square (RMS) value, written as ${\displaystyle V_{\rm {rms}}}$, because

${\displaystyle P_{\rm {time~averaged}}={\frac {{V_{\rm {rms}}}^{2}}{R}}.}$

Power oscillation

${\displaystyle v(t)=V_{\mathrm {peak} }\sin(\omega t)}$

${\displaystyle i(t)={\frac {v(t)}{R}}={\frac {V_{\mathrm {peak} }}{R}}\sin(\omega t)}$

${\displaystyle P(t)=v(t)\ i(t)={\frac {(V_{\mathrm {peak} })^{2}}{R}}\sin ^{2}(\omega t)}$

Root mean square voltage

Below it is assumed an AC waveform (with no DC component).
The RMS voltage is the square Root of the Mean over one cycle of the Square of the instantaneous voltage.

• For an arbitrary periodic waveform ${\displaystyle v(t)}$ of period ${\displaystyle T}$:

${\displaystyle V_{\mathrm {rms} }={\sqrt {{\frac {1}{T}}\int _{0}^{T}{[v(t)]^{2}dt}}}.}$

• For a sinusoidal voltage:

${\displaystyle {\begin{aligned}V_{\mathrm {rms} }&={\sqrt {{\frac {1}{T}}\int _{0}^{T}[{V_{pk}\sin(\omega t+\phi )]^{2}dt}}}\\&=V_{pk}{\sqrt {{\frac {1}{2T}}\int _{0}^{T}[{1-\cos(2\omega t+2\phi )]dt}}}\\&=V_{pk}{\sqrt {{\frac {1}{2T}}\int _{0}^{T}{dt}}}\\&={\frac {V_{pk}}{\sqrt {2}}}\end{aligned}}}$

where the trigonometric identity ${\displaystyle \sin ^{2}x={\frac {1-\cos 2x}{2}}}$ has been used and the factor ${\displaystyle {\sqrt {2}}}$ is called the crest factor, which varies for different waveforms.

• For a triangle waveform centered about zero

${\displaystyle V_{\mathrm {rms} }={\frac {V_{\mathrm {peak} }}{\sqrt {3}}}.}$

• For a square waveform centered about zero

${\displaystyle \displaystyle V_{\mathrm {rms} }=V_{\mathrm {peak} }.}$

Example

To illustrate these concepts, consider a 230 V AC mains supply used in many countries around the world. It is so called because its root mean square value is 230 V. This means that the time-averaged power delivered 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:

${\displaystyle V_{\mathrm {peak} }={\sqrt {2}}\ V_{\mathrm {rms} }.}$

For 230 V AC, the peak voltage ${\displaystyle V_{\mathrm {peak} }}$ is therefore ${\displaystyle 230V\times {\sqrt {2}}}$, which is about 325 V. During the course of one cycle the voltage rises from zero to 325 V, falls through zero to -325 V, and returns to zero.

Information transmission

Alternating current is used to transmit information, as in the cases of telephone and cable television. Information signals are carried over a wide range of AC frequencies. POTS telephone signals have a frequency of about 3 kHz, close to the baseband 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 over the air.

History

The first alternator to produce alternating current was a dynamo electric generator based on Michael Faraday's principles constructed by the French instrument maker Hippolyte Pixii in 1832. Pixii later added a commutator to his device to produce the (then) more commonly used direct current. The earliest recorded practical application of alternating current is by Guillaume Duchenne, inventor and developer of electrotherapy. In 1855, he announced that AC was superior to direct current for electrotherapeutic triggering of muscle contractions. Alternating current technology had first developed in Europe due to the work of Guillaume Duchenne (1850s), the Hungarian Ganz Works company (1870s), and in the 1880s: Sebastian Ziani de Ferranti, Lucien Gaulard, and Galileo Ferraris.

In 1876, Russian engineer Pavel Yablochkov 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 'electric candles' (arc lamps) of his own design, used to keep the failure of one lamp from disabling the entire circuit. In 1878, the Ganz factory, Budapest, Hungary, began manufacturing equipment for electric lighting and, by 1883, had installed over fifty systems in Austria-Hungary. Their AC systems used arc and incandescent lamps, generators, and other equipment.

Transformers

Alternating current systems can use transformers to change voltage from low to high level and back, allowing generation and consumption at low voltages but transmission, possibly over great distances, at high voltage, with savings in the cost of conductors and energy losses. A bipolar open-core power transformer developed by Lucien Gaulard and John Dixon Gibbs was demonstrated in London in 1881, and attracted the interest of Westinghouse. They also exhibited the invention in Turin in 1884. However these early induction coils with open magnetic circuits are inefficient at transferring power to loads. 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. The direct current systems did not have these drawbacks, giving it significant advantages over early AC systems.

Pioneers

The Hungarian "ZBD" Team (Károly Zipernowsky, Ottó Bláthy, Miksa Déri), inventors of the first high efficiency, closed-core shunt connection transformer

 

The prototype of the ZBD transformer on display at the Széchenyi István Memorial Exhibition, Nagycenk in Hungary

In the autumn of 1884, Károly Zipernowsky, Ottó Bláthy and Miksa Déri (ZBD), three engineers associated with the Ganz factory, determined that open-core devices were impractical, as they were incapable of reliably regulating voltage. 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 a wound around iron wire ring core or b) surrounded by iron wire core. 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 toroidal cores). The new transformers were 3.4 times more efficient than the open-core bipolar devices of Gaulard and Gibbs. The Ganz factory in 1884 shipped the world's first five high-efficiency AC transformers. 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.

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 1400 V to 2000 V) than the voltage of utilization loads (100 V initially preferred). 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. The other essential milestone was the introduction of 'voltage source, voltage intensive' (VSVI) systems' by the invention of constant voltage generators in 1885. Ottó Bláthy also invented the first AC electricity meter.

The AC power systems was developed and adopted rapidly after 1886 due to its ability to distribute electricity efficiently over long distances, overcoming the limitations of the direct current system. In 1886, the ZBD engineers designed the world's first power station that used AC generators to power a parallel-connected common electrical network, the steam-powered Rome-Cerchi power plant. The reliability of the AC technology received impetus after the Ganz Works electrified a large European metropolis: Rome in 1886.

Westinghouse Early AC System 1887
(US patent 373035)

In the UK, Sebastian de Ferranti, who had been developing AC generators and transformers in London since 1882, redesigned the AC system at the Grosvenor Gallery power station in 1886 for the London Electric Supply Corporation (LESCo) including alternators of his own design and transformer designs similar to Gaulard and Gibbs. In 1890 he designed their power station at Deptford and converted the Grosvenor Gallery station across the Thames into an electrical substation, showing the way to integrate older plants into a universal AC supply system.

In the US William Stanley, Jr. designed one of the first practical devices to transfer AC power efficiently between isolated circuits. Using pairs of coils wound on a common iron core, his design, called an induction coil, was an early (1885) transformer. Stanley also worked on engineering and adapting European designs such as the Gaulard and Gibbs transformer for US entrepreneur George Westinghouse who started building AC systems in 1886. The spread of Westinghouse and other AC systems triggered a push back in late 1887 by Edison (a proponent of direct current) who attempted to discredit alternating current as too dangerous in a public campaign called the "War of Currents". In 1888 alternating current systems gained further viability with introduction of a functional AC motor, something these systems had lacked up till then. The design, an induction motor, was independently invented by Galileo Ferraris and Nikola Tesla (with Tesla's design being licensed by Westinghouse in the US). This design was further developed into the modern practical three-phase form by Mikhail Dolivo-Dobrovolsky and Charles Eugene Lancelot Brown.

The Ames Hydroelectric Generating Plant (spring of 1891) and the original Niagara Falls Adams Power Plant (August 25, 1895) were among the first hydroelectric alternating current power plants. The first long distance transmission of single-phase electricity was from a hydroelectric generating plant in Oregon at Willamette Falls which in 1890 sent power fourteen miles downriver to downtown Portland for street lighting. In 1891, a second transmission system was installed in Telluride Colorado. 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 Almarian William Decker to provide power to the city of Pomona, California which was 14 miles away. In 1893 he next designed the first commercial three-phase power plant in the United States using alternating current was the hydroelectric Mill Creek No. 1 Hydroelectric Plant near Redlands, California. Decker's design incorporated 10 kV three-phase transmission and established the standards for the complete system of generation, transmission and motors used today. The Jaruga Hydroelectric Power Plant in Croatia was set in operation on 28 August 1895. The two generators (42 Hz, 550 kW each) and the transformers were produced and installed by the Hungarian company Ganz. The transmission line from the power plant to the City of Šibenik was 11.5 kilometers (7.1 mi) long on wooden towers, and the municipal distribution grid 3000 V/110 V included six transforming stations. 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 Charles Steinmetz, Oliver Heaviside, and many others. Calculations in unbalanced three-phase systems were simplified by the symmetrical components methods discussed by Charles Legeyt Fortescue in 1918.