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Power factor
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For the firearms cartridge ranking system, see Power factor (shooting sports).
In electrical engineering, the power factor of an AC electrical power system is defined as the ratio of the real power absorbed by the load to the apparent power flowing in the circuit, and is a dimensionless number in the closed interval of −1 to 1. A power factor of less than one indicates the voltage and current are not in phase, reducing the instantaneous product of the two. Real power is the instantaneous product of voltage and current and represents the capacity of the electricity for performing work. Apparent power is the product of average current and voltage. Due to energy stored in the load and returned to the source, or due to a non-linear load that distorts the wave shape of the current drawn from the source, the apparent power may be greater than the real power. A negative power factor occurs when the device (which is normally the load) generates power, which then flows back towards the source.
In an electric power system, a load with a low power factor draws more current than a load with a high power factor for the same amount of useful power transferred. The higher currents increase the energy lost in the distribution system, and require larger wires and other equipment. Because of the costs of larger equipment and wasted energy, electrical utilities will usually charge a higher cost to industrial or commercial customers where there is a low power factor.
Power-factor correction increases the power factor of a load, improving efficiency for the distribution system to which it is attached. Linear loads with low power factor (such as induction motors) can be corrected with a passive network of capacitors or inductors. Non-linear loads, such as rectifiers, distort the current drawn from the system. In such cases, active or passive power factor correction may be used to counteract the distortion and raise the power factor. The devices for correction of the power factor may be at a central substation, spread out over a distribution system, or built into power-consuming equipment.
Contents
1 Linear circuits
1.1 Definition and calculation
1.1.1 Power triangle
1.1.2 Increasing the power factor
1.1.3 Decreasing the power factor
1.1.4 Lagging and leading power factors
1.2 Power factor correction of linear loads
2 Non-linear loads
2.1 Non-sinusoidal components
2.2 Distortion power factor
2.3 Distortion in three-phase networks
2.4 Switched-mode power supplies
2.5 Power factor correction (PFC) in non-linear loads
2.5.1 Passive PFC
2.5.2 Active PFC
2.5.3 Dynamic PFC
3 Importance of power factor in distribution systems
4 Techniques for measuring the power factor
5 Mnemonics
6 References
7 External links
Linear circuits[edit]
Instantaneous and average power calculated from AC voltage and current with a zero power factor ( φ = 90 ∘ {\displaystyle \varphi =90^{\circ }} , cos  φ = 0 {\displaystyle \cos \varphi =0} ). The blue line shows all the power is stored temporarily in the load during the first quarter cycle and returned to the grid during the second quarter cycle, so no real power is consumed.
Instantaneous and average power calculated from AC voltage and current with a lagging power factor ( φ = 45 ∘ {\displaystyle \varphi =45^{\circ }} , cos  φ ≈ 0.71 {\displaystyle \cos \varphi \approx 0.71} ). The blue line shows some of the power is returned to the grid during the part of the cycle labeled φ {\displaystyle \varphi } .
In a purely resistive AC circuit, voltage and current waveforms are in step (or in phase), changing polarity at the same instant in each cycle. All the power entering the load is consumed (or dissipated).
Where reactive loads are present, such as with capacitors or inductors, energy storage in the loads results in a phase difference between the current and voltage waveforms. During each cycle of the AC voltage, extra energy, in addition to any energy consumed in the load, is temporarily stored in the load in electric or magnetic fields, and then returned to the power grid a fraction of the period later.
Because high voltage alternating current distribution systems are essentially quasi-linear circuit systems subject to continuous daily variation, there is a continuous "ebb and flow" of nonproductive power. Non productive power increases the current in the line, potentially to the point of failure.
Thus, a circuit with a low power factor will use higher currents to transfer a given quantity of real power than a circuit with a high power factor. A linear load does not change the shape of the waveform of the current, but may change the relative timing (phase) between voltage and current.
Electrical circuits containing dominantly resistive loads (incandescent lamps, heating elements) have a power factor of almost 1.0, but circuits containing inductive or capacitive loads (electric motors, solenoid valves, transformers, fluorescent lamp ballasts, and others) can have a power factor well below 1.
Definition and calculation[edit]
AC power flow has two components:
Real power or active power ( P {\displaystyle P} ) (sometimes called average power[1]), expressed in watts (W)
Reactive power ( Q {\displaystyle Q} ), usually expressed in reactive volt-amperes (var)[2]
These are combined to the complex power ( S {\displaystyle S} ) expressed volt-amperes (VA). The magnitude of the complex power is the apparent power ( | S | {\displaystyle |S|} ), also expressed in volt-amperes (VA).
The VA and var are non-SI units mathematically identical to the watt, but are used in engineering practice instead of the watt to state what quantity is being expressed. The SI explicitly disallows using units for this purpose or as the only source of information about a physical quantity as used.[3]
The power factor is defined as the ratio of real power to apparent power. As power is transferred along a transmission line, it does not consist purely of real power that can do work once transferred to the load, but rather consists of a combination of real and reactive power, called apparent power. The power factor describes the amount of real power transmitted along a transmission line relative to the total apparent power flowing in the line.[4][5]
Power triangle[edit]
One can relate the various components of AC power by using the power triangle in vector space. Real power extends horizontally in the î direction as it represents a purely real component of AC power. Reactive power extends in the direction of ĵ as it represents a purely imaginary component of AC power. Complex power (and its magnitude, Apparent power) represents a combination of both real and reactive power, and therefore can be calculated by using the vector sum of these two components. We can conclude that the mathematical relationship between these components is:
S = P + j Q | S | 2 = P 2 + Q 2 | S | = P 2 + Q 2 cos  θ , power factor = P , real power | S | , apparent power {\displaystyle {\begin{aligned}S&=P+jQ\\|S|^{2}&=P^{2}+Q^{2}\\[3pt]|S|&={\sqrt {P^{2}+Q^{2}}}\\\cos \theta {\text{, power factor}}&={\frac {P{\text{, real power}}}{|S|{\text{, apparent power}}}}\end{aligned}}}
Increasing the power factor[edit]
As the power factor (i.e. cos θ) increases, the ratio of real power to apparent power (which = cos θ), increases and approaches unity (1), while the angle θ decreases and the reactive power decreases. [As cos θ → 1, its maximum possible value, θ → 0 and so Q → 0, as the load becomes less reactive and more purely resistive].
Decreasing the power factor[edit]
As the power factor decreases, the ratio of real power to apparent power also decreases, as the angle θ increases and reactive power increases.
Lagging and leading power factors[edit]
There is also a difference between a lagging and leading power factor. The terms refer to whether the phase of the current is leading or lagging the phase of the voltage. A lagging power factor signifies that the load is inductive, as the load will “consume” reactive power, and therefore the reactive component Q {\displaystyle Q} is positive as reactive power travels through the circuit and is “consumed” by the inductive load. A leading power factor signifies that the load is capacitive, as the load “supplies” reactive power, and therefore the reactive component Q {\displaystyle Q} is negative as reactive power is being supplied to the circuit.
If θ is the phase angle between the current and voltage, then the power factor is equal to the cosine of the angle, cos  θ {\displaystyle \cos \theta } :
| P | = | S | cos  θ {\displaystyle |P|=|S|\cos \theta }
Since the units are consistent, the power factor is by definition a dimensionless number between −1 and 1. When power factor is equal to 0, the energy flow is entirely reactive and stored energy in the load returns to the source on each cycle. When the power factor is 1, all the energy supplied by the source is consumed by the load. Power factors are usually stated as "leading" or "lagging" to show the sign of the phase angle. Capacitive loads are leading (current leads voltage), and inductive loads are lagging (current lags voltage).
If a purely resistive load is connected to a power supply, current and voltage will change polarity in step, the power factor will be 1, and the electrical energy flows in a single direction across the network in each cycle. Inductive loads such as induction motors (any type of wound coil) consume reactive power with current waveform lagging the voltage. Capacitive loads such as capacitor banks or buried cable generate reactive power with current phase leading the voltage. Both types of loads will absorb energy during part of the AC cycle, which is stored in the device's magnetic or electric field, only to return this energy back to the source during the rest of the cycle.
For example, to get 1 kW of real power, if the power factor is unity, 1 kVA of apparent power needs to be transferred (1 kW ÷ 1 = 1 kVA). At low values of power factor, more apparent power needs to be transferred to get the same real power. To get 1 kW of real power at 0.2 power factor, 5 kVA of apparent power needs to be transferred (1 kW ÷ 0.2 = 5 kVA). This apparent power must be produced and transmitted to the load, and is subject to the losses in the production and transmission processes.
Electrical loads consuming alternating current power consume both real power and reactive power. The vector sum of real and reactive power is the apparent power. The presence of reactive power causes the real power to be less than the apparent power, and so, the electric load has a power factor of less than 1.
A negative power factor (0 to −1) can result from returning power to the source, such as in the case of a building fitted with solar panels when surplus power is fed back into the supply.[6][7][8]
Power factor correction of linear loads[edit]
A high power factor is generally desirable in a power delivery system to reduce losses and improve voltage regulation at the load. Compensating elements near an electrical load will reduce the apparent power demand on the supply system. Power factor correction may be applied by an electric power transmission utility to improve the stability and efficiency of the network. Individual electrical customers who are charged by their utility for low power factor may install correction equipment to increase their power factor so as to reduce costs.
Power factor correction brings the power factor of an AC power circuit closer to 1 by supplying or absorbing reactive power, adding capacitors or inductors that act to cancel the inductive or capacitive effects of the load, respectively. In the case of offsetting the inductive effect of motor loads, capacitors can be locally connected. These capacitors help to generate reactive power to meet the demand of the inductive loads. This will keep that reactive power from having to flow all the way from the utility generator to the load. In the electricity industry, inductors are said to consume reactive power and capacitors are said to supply it, even though reactive power is just energy moving back and forth on each AC cycle.
The reactive elements in power factor correction devices can create voltage fluctuations and harmonic noise when switched on or off. They will supply or sink reactive power regardless of whether there is a corresponding load operating nearby, increasing the system's no-load losses. In the worst case, reactive elements can interact with the system and with each other to create resonant conditions, resulting in system instability and severe overvoltage fluctuations. As such, reactive elements cannot simply be applied without engineering analysis.
1. Reactive Power Control Relay; 2. Network connection points; 3. Slow-blow Fuses; 4. Inrush Limiting Contactors; 5. Capacitors (single-phase or three-phase units, delta-connection); 6. Transformer (for controls and ventilation fans)
An automatic power factor correction unit consists of a number of capacitors that are switched by means of contactors. These contactors are controlled by a regulator that measures power factor in an electrical network. Depending on the load and power factor of the network, the power factor controller will switch the necessary blocks of capacitors in steps to make sure the power factor stays above a selected value.
Instead of using a set of switched capacitors, an unloaded synchronous motor can supply reactive power. The reactive power drawn by the synchronous motor is a function of its field excitation. This is referred to as a synchronous condenser. It is started and connected to the electrical network. It operates at a leading power factor and puts vars onto the network as required to support a system's voltage or to maintain the system power factor at a specified level.
The synchronous condenser's installation and operation are identical to large electric motors. Its principal advantage is the ease with which the amount of correction can be adjusted; it behaves like a variable capacitor. Unlike capacitors, the amount of reactive power supplied is proportional to voltage, not the square of voltage; this improves voltage stability on large networks. Synchronous condensers are often used in connection with high-voltage direct-current transmission projects or in large industrial plants such as steel mills.
For power factor correction of high-voltage power systems or large, fluctuating industrial loads, power electronic devices such as the Static VAR compensator or STATCOM are increasingly used. These systems are able to compensate sudden changes of power factor much more rapidly than contactor-switched capacitor banks, and being solid-state require less maintenance than synchronous condensers.
Non-linear loads[edit]
Examples of non-linear loads on a power system are rectifiers (such as used in a power supply), and arc discharge devices such as fluorescent lamps, electric welding machines, or arc furnaces. Because current in these systems is interrupted by a switching action, the current contains frequency components that are multiples of the power system frequency. Distortion power factor is a measure of how much the harmonic distortion of a load current decreases the average power transferred to the load.
Sinusoidal voltage and non-sinusoidal current give a distortion power factor of 0.75 for this computer power supply load.
Non-sinusoidal components[edit]
In linear circuits having only sinusoidal currents and voltages of one frequency, the power factor arises only from the difference in phase between the current and voltage. This is "displacement power factor".[9]
Non-linear loads change the shape of the current waveform from a sine wave to some other form. Non-linear loads create harmonic currents in addition to the original (fundamental frequency) AC current. This is of importance in practical power systems that contain non-linear loads such as rectifiers, some forms of electric lighting, electric arc furnaces, welding equipment, switched-mode power supplies, variable speed drives and other devices. Filters consisting of linear capacitors and inductors can prevent harmonic currents from entering the supplying system.
To measure the real power or reactive power, a wattmeter designed to work properly with non-sinusoidal currents must be used.
Distortion power factor[edit]
The distortion power factor is the distortion component associated with the harmonic voltages and currents present in the system.
distortion power factor = 1 1 + THD i 2 = I 1, rms I rms {\displaystyle {\mbox{distortion power factor}}={1 \over {\sqrt {1+{\mbox{THD}}_{i}^{2}}}}={I_{\mbox{1, rms}} \over I_{\mbox{rms}}}}
THD i {\displaystyle {\mbox{THD}}_{i}} is the total harmonic distortion of the load current. I 1 , rms {\displaystyle I_{1,{\mbox{rms}}}} is the fundamental component of the current and I rms {\displaystyle I_{\mbox{rms}}} is the total current – both are root mean square-values (distortion power factor can also be used to describe individual order harmonics, using the corresponding current in place of total current). This definition with respect to total harmonic distortion assumes that the voltage stays undistorted (sinusoidal, without harmonics). This simplification is often a good approximation for stiff voltage sources (not being affected by changes in load downstream in the distribution network). Total harmonic distortion of typical generators from current distortion in the network is on the order of 1–2%, which can have larger scale implications but can be ignored in common practice.[10]
The result when multiplied with the displacement power factor (DPF) is the overall, true power factor or just power factor (PF):
PF = I 1, rms I rms cos  φ {\displaystyle {\mbox{PF}}={I_{\mbox{1, rms}} \over I_{\mbox{rms}}}\cos \varphi }
Distortion in three-phase networks[edit]
In practice, the local effects of distortion current on devices in a three-phase distribution network rely on the magnitude of certain order harmonics rather than the total harmonic distortion.
For example, the triplen, or zero-sequence, harmonics (3rd, 9th, 15th, etc.) have the property of being in-phase when compared line-to-line. In a delta-wye transformer, these harmonics can result in circulating currents in the delta windings and result in greater resistive heating. In a wye-configuration of a transformer, triplen harmonics will not create these currents, but they will result in a non-zero current in the neutral wire. This could overload the neutral wire in some cases and create error in kilowatt-hour metering systems and billing revenue.[11][12] The presence of current harmonics in a transformer also result in larger eddy currents in the magnetic core of the transformer. Eddy current losses generally increase as the square of the frequency, lowering the transformer's efficiency, dissipating additional heat, and reducing its service life.[13]
Negative-sequence harmonics (5th, 11th, 17th, etc.) combine 120 degrees out of phase, similarly to the fundamental harmonic but in a reversed sequence. In generators and motors, these currents produce magnetic fields which oppose the rotation of the shaft and sometimes result in damaging mechanical vibrations.[14]
Switched-mode power supplies[edit]
Main article: switched-mode power supply § Power factor
A particularly important class of non-linear loads is the millions of personal computers that typically incorporate switched-mode power supplies (SMPS) with rated output power ranging from a few watts to more than 1 kW. Historically, these very-low-cost power supplies incorporated a simple full-wave rectifier that conducted only when the mains instantaneous voltage exceeded the voltage on the input capacitors. This leads to very high ratios of peak-to-average input current, which also lead to a low distortion power factor and potentially serious phase and neutral loading concerns.
A typical switched-mode power supply first converts the AC mains to a DC bus by means of a bridge rectifier or a similar circuit. The output voltage is then derived from this DC bus. The problem with this is that the rectifier is a non-linear device, so the input current is highly non-linear. That means that the input current has energy at harmonics of the frequency of the voltage.
This presents a particular problem for the power companies, because they cannot compensate for the harmonic current by adding simple capacitors or inductors, as they could for the reactive power drawn by a linear load. Many jurisdictions are beginning to legally require power factor correction for all power supplies above a certain power level.
Regulatory agencies such as the EU have set harmonic limits as a method of improving power factor. Declining component cost has hastened implementation of two different methods. To comply with current EU standard EN61000-3-2, all switched-mode power supplies with output power more than 75 W must include passive power factor correction, at least. 80 Plus power supply certification requires a power factor of 0.9 or more.[15]
Power factor correction (PFC) in non-linear loads[edit]
Passive PFC[edit]
The simplest way to control the harmonic current is to use a filter that passes current only at line frequency (50 or 60 Hz). The filter consists of capacitors or inductors, and makes a non-linear device look more like a linear load. An example of passive PFC is a valley-fill circuit.
A disadvantage of passive PFC is that it requires larger inductors or capacitors than an equivalent power active PFC circuit.[16][17][18] Also, in practice, passive PFC is often less effective at improving the power factor.[19][20][21][22][23]
Active PFC[edit]
Specifications taken from the packaging of a 610 W PC power supply showing active PFC rating
Active PFC is the use of power electronics to change the waveform of current drawn by a load to improve the power factor.[24] Some types of the active PFC are buck, boost, buck-boost and synchronous condenser. Active power factor correction can be single-stage or multi-stage.
In the case of a switched-mode power supply, a boost converter is inserted between the bridge rectifier and the main input capacitors. The boost converter attempts to maintain a constant DC bus voltage on its output while drawing a current that is always in phase with and at the same frequency as the line voltage. Another switched-mode converter inside the power supply produces the desired output voltage from the DC bus. This approach requires additional semiconductor switches and control electronics, but permits cheaper and smaller passive components. It is frequently used in practice.
For a three-phase SMPS, the Vienna rectifier configuration may be used to substantially improve the power factor.
SMPSs with passive PFC can achieve power factor of about 0.7–0.75, SMPSs with active PFC, up to 0.99 power factor, while a SMPS without any power factor correction have a power factor of only about 0.55–0.65.[25]
Due to their very wide input voltage range, many power supplies with active PFC can automatically adjust to operate on AC power from about 100 V (Japan) to 240 V (Europe). That feature is particularly welcome in power supplies for laptops.
Dynamic PFC[edit]
Dynamic power factor correction (DPFC), sometimes referred to as "real-time power factor correction," is used for electrical stabilization in cases of rapid load changes (e.g. at large manufacturing sites). DPFC is useful when standard power factor correction would cause over or under correction.[26] DPFC uses semiconductor switches, typically thyristors, to quickly connect and disconnect capacitors or inductors to improve power factor.
Importance of power factor in distribution systems[edit]
75 Mvar capacitor bank in a 150 kV substation
Power factors below 1.0 require a utility to generate more than the minimum volt-amperes necessary to supply the real power (watts). This increases generation and transmission costs. For example, if the load power factor were as low as 0.7, the apparent power would be 1.4 times the real power used by the load. Line current in the circuit would also be 1.4 times the current required at 1.0 power factor, so the losses in the circuit would be doubled (since they are proportional to the square of the current). Alternatively all components of the system such as generators, conductors, transformers, and switchgear would be increased in size (and cost) to carry the extra current.
Utilities typically charge additional costs to commercial customers who have a power factor below some limit, which is typically 0.9 to 0.95. Engineers are often interested in the power factor of a load as one of the factors that affect the efficiency of power transmission.
With the rising cost of energy and concerns over the efficient delivery of power, active PFC has become more common in consumer electronics.[27] Current Energy Star guidelines for computers[28] call for a power factor of ≥ 0.9 at 100% of rated output in the PC's power supply. According to a white paper authored by Intel and the U.S. Environmental Protection Agency, PCs with internal power supplies will require the use of active power factor correction to meet the ENERGY STAR 5.0 Program Requirements for Computers.[29]
In Europe, EN 61000-3-2 requires power factor correction be incorporated into consumer products.
Small customers, such as households, are not usually charged for reactive power and so power factor metering equipment for such customers will not be installed.
Techniques for measuring the power factor[edit]
The power factor in a single-phase circuit (or balanced three-phase circuit) can be measured with the wattmeter-ammeter-voltmeter method, where the power in watts is divided by the product of measured voltage and current. The power factor of a balanced polyphase circuit is the same as that of any phase. The power factor of an unbalanced poly phase circuit is not uniquely defined.
A direct reading power factor meter can be made with a moving coil meter of the electrodynamic type, carrying two perpendicular coils on the moving part of the instrument. The field of the instrument is energized by the circuit current flow. The two moving coils, A and B, are connected in parallel with the circuit load. One coil, A, will be connected through a resistor and the second coil, B, through an inductor, so that the current in coil B is delayed with respect to current in A. At unity power factor, the current in A is in phase with the circuit current, and coil A provides maximum torque, driving the instrument pointer toward the 1.0 mark on the scale. At zero power factor, the current in coil B is in phase with circuit current, and coil B provides torque to drive the pointer towards 0. At intermediate values of power factor, the torques provided by the two coils add and the pointer takes up intermediate positions.[30]
Another electromechanical instrument is the polarized-vane type.[31] In this instrument a stationary field coil produces a rotating magnetic field, just like a polyphase motor. The field coils are connected either directly to polyphase voltage sources or to a phase-shifting reactor if a single-phase application. A second stationary field coil, perpendicular to the voltage coils, carries a current proportional to current in one phase of the circuit. The moving system of the instrument consists of two vanes that are magnetized by the current coil. In operation the moving vanes take up a physical angle equivalent to the electrical angle between the voltage source and the current source. This type of instrument can be made to register for currents in both directions, giving a four-quadrant display of power factor or phase angle.
Digital instruments exist that directly measure the time lag between voltage and current waveforms. Low-cost instruments of this type measure the peak of the waveforms. More sophisticated versions measure the peak of the fundamental harmonic only, thus giving a more accurate reading for phase angle on distorted waveforms. Calculating power factor from voltage and current phases is only accurate if both waveforms are sinusoidal.[32]
Power Quality Analyzers, often referred to as Power Analyzers, make a digital recording of the voltage and current waveform (typically either one phase or three phase) and accurately calculate true power (watts), apparent power (VA) power factor, AC voltage, AC current, DC voltage, DC current, frequency, IEC61000-3-2/3-12 Harmonic measurement, IEC61000-3-3/3-11 flicker measurement, individual phase voltages in delta applications where there is no neutral line, total harmonic distortion, phase and amplitude of individual voltage or current harmonics, etc.[33][34]
Mnemonics[edit]
English-language power engineering students are advised to remember: "ELI the ICE man" or "ELI on ICE" – the voltage E leads the current I in an inductor L; the current leads the voltage in a capacitor C.
Another common mnemonic is CIVIL – in a capacitor (C) the current (I) leads voltage (V), voltage (V) leads current (I) in an inductor (L).
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Jump up ^ Manual of Electric Instruments Construction and Operating Principles, Schenectady, New York: General Electric, Meter and Instrument Department, 1949, pp. 66–68, GET-1087A
Jump up ^ "The Fundamentals of FFT-Based Signal Analysis and Measurement in LabVIEW and LabWindows/CVI". National Instruments Corporation. Retrieved 6 November 2017.
Jump up ^ "WT3000E Series Precision Power Analyzers" (PDF). Yokogawa Corporation. Retrieved 6 November 2017.
Jump up ^ "Fluke 1760 Three-Phase Power Quality Recorder" (PDF). Fluke Corporation. Retrieved 6 November 2017.
External links[edit]
Harmonics and how they relate to power factor (PDF), U Texas .
NIST Team Demystifies Utility of Power Factor Correction Devices, NIST, December 15, 2009 .
Power factor calculation and correction, US .
Retrieved from "https://en.wikipedia.org/w/index.php?title=Power_factor&oldid=865680792"
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text:
Let's Spend the Night Together
From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
For the 1982 Rolling Stones concert film, see Let's Spend the Night Together (film).
"Let's Spend the Night Together"
UK single picture sleeve
Single by the Rolling Stones
A-side
"Ruby Tuesday" (double A-side)
Released
January 1967 (1967-01)
Format
7-inch single
Recorded
December 1966
Studio
RCA, Hollywood, California
Genre
Rock
Length
3:29
Label
Decca (UK)
London (US)
Songwriter(s)
Jagger/Richards
Producer(s)
Andrew Loog Oldham
The Rolling Stones singles chronology
"Have You Seen Your Mother, Baby, Standing in the Shadow?"
(1966) "Let's Spend the Night Together" / "Ruby Tuesday"
(1967) "We Love You" / "Dandelion"
(1967)
Audio
"Let's Spend the Night Together" on YouTube
Alternative cover
French single picture sleeve
"Let's Spend the Night Together" is a song written by Mick Jagger and Keith Richards, and originally released by the Rolling Stones as a double A-sided single together with "Ruby Tuesday" in January 1967.[1] It also appears as the opening track on the American version of their album Between the Buttons. It has been covered by various artists, including David Bowie in 1973.[2]
Contents
1 Recording
2 Original release
2.1 Personnel
2.2 Charts
2.3 Other releases
2.4 Live performances
3 David Bowie version
3.1 Track listing
3.2 Personnel
3.3 Live versions
3.4 Other releases
4 See also
5 References
6 External links
Recording[edit]
The song was recorded in December 1966 at the RCA Records studio in Hollywood, California, where the group recorded most of their 1965–1966 hits.[3] Recording engineer Glyn Johns recounts that while mixing "Let's Spend the Night Together", Oldham was trying to get a certain sound by clicking his fingers.[4] Two policemen showed up, stating that the front door was open and that they were checking to see if everything was all right. At first, Oldham asked them to hold his earphones while he snapped his fingers but then Johns said they needed a more wooden sound. The policemen suggested their truncheons and Mick Jagger took the truncheons into the studio to record the claves-like sound that can be heard during the quiet break at one minute 40 seconds into the song.
Original release[edit]
Released in the United Kingdom as a single on 13 January 1967, "Let's Spend the Night Together" reached number three on the UK Singles Chart, while the other side of the single, "Ruby Tuesday" was not listed. In the United States, the single was released in January[3] and became the opening track of the American edition of the Stones' album Between the Buttons. Both songs entered the Billboard Hot 100 singles chart on 21 January.[5] However, by 4 March, "Ruby Tuesday" reached number one, while "Let's Spend the Night Together" stalled at number 55.[6] Due to the sexually charged nature of the lyrics, "Let's Spend the Night Together" received less airplay.[2] In the Cash Box chart, which was based on sales only, the song reached number 28.[7] In other countries worldwide, both sides of the single charted separately. In Ireland for example, "Ruby Tuesday" peaked at number six, while "Let's Spend The Night Together" charted separately at number 14, as Ireland's national broadcaster, RTÉ, considered "Ruby Tuesday" to be more suitable for radio airplay.
The song features piano by Rolling Stones contributor Jack Nitzsche,[3] organ by Brian Jones, drums by Charlie Watts, piano, electric guitar and bass by Richards, lead vocals by Jagger and backing vocals from both Jagger and Richards. Usual bassist Bill Wyman does not appear on the recording.
On The Ed Sullivan Show, the band was initially refused permission to perform the number. Sullivan himself told Jagger, "Either the song goes or you go".[8] A compromise was reached to substitute the words "let's spend some time together" in place of "let's spend the night together"; Jagger agreed to change the lyrics but ostentatiously rolled his eyes at the TV camera while singing them, as well as Bill Wyman. When the Rolling Stones, following their performance of the song, returned on stage, they were all dressed up in Nazi uniforms with swastikas, which caused Sullivan to angrily order them to return to their dressing rooms to change back into their performance clothes, at which they left the studio altogether.[9] As a result of this incident, Sullivan announced that the Rolling Stones would be banned from performing on his show again.[9] However, the Stones did appear on the show again and performed three songs on 23 November 1969.[10] In April 2006, for their first-ever performance in China, authorities prohibited the group from performing the song due to its "suggestive lyrics".[11]
Personnel[edit]
Mick Jagger – lead vocal, backing vocals, truncheons, percussion
Keith Richards – electric guitars, bass, piano, backing vocals
Brian Jones – organ, electric guitar
Charlie Watts – drums
Jack Nitzsche – piano[12]
Charts[edit]
Chart (1967)
Peak
position
Austria (Ö3 Austria Top 40)[13] 3
Belgium (Ultratop 50 Flanders)[14] 7
Germany (Official German Charts)[15] 1
Ireland (IRMA)[16] 14
Norway (VG-lista)[17] 2
UK Singles (Official Charts Company)[18] 3
US Billboard Hot 100[19] 55
US Cash Box Top 100[20] 28
Other releases[edit]
"Let's Spend the Night Together" was released on the US edition of the Stones' 1967 studio album Between the Buttons and on the following compilation albums:
Flowers (1967)
Through the Past, Darkly (Big Hits Vol. 2) (1969)
Hot Rocks 1964–1971 (1971)
Rolled Gold: The Very Best of the Rolling Stones (1975)
30 Greatest Hits (1977)
Singles Collection: The London Years (1989)
Forty Licks (2002)
Singles 1965–1967 (2004)
GRRR! (2012)
A live version appeared on Still Life (1982).
Live performances[edit]
After performing the song on a 28-date European tour in spring 1967, it was not until 1976 that The Rolling Stones next played "Let's Spend the Night Together" live in concert regularly. A one-off performance of the song was played at the Knebworth Festival in 1976 and it also followed along and was played a few times in 1977. It was performed regularly on the 1981 and 1982 tours before being again retired for 15 years where it was played occasionally during the Bridges To Babylon Tour in 1997 and 1998. It was then absent during the No Security Tour 1999 and the Licks Tour 2002–2003. The song was occasionally played at most shows and became a minor mainstay of their live performances during their two-year A Bigger Bang Tour. The song has been played a few times during the band's 14 On Fire tour.
1967 European tour
1977 El Mocambo Club, Toronto
1981 American tour
1982 European tour
1997 North American tour
1998 Spring Tour (North America), European tour
2005–2006 US & Canada
2006 Asia & Australasia, Europe, US & Canada
2007 European tour
2014 European tour
2015 American tour
2016 Latin America Olé tour
David Bowie version[edit]
"Let's Spend the Night Together"
Single by David Bowie
from the album Aladdin Sane
B-side
"Lady Grinning Soul"
Released
July 1973 (1973-07)
Format
7-inch single
Recorded
December 1972 or January 1973
Studio
RCA, New York City, or Trident, London
Genre
Glam rock
Length
3:03
Label
RCA
Songwriter(s)
Jagger/Richards
Producer(s)
Ken Scott
David Bowie
David Bowie singles chronology
"Time"
(1973) "Let's Spend the Night Together"
(1973) "Life on Mars?"
(1973)
David Bowie recorded a glam rock cover version of "Let's Spend the Night Together" for his Aladdin Sane album, released in April 1973. It was also issued as a single by RCA Records in the US and Europe. It was a Dutch Top 40 hit, peaking at number 19[21]
Bowie's rendition featured pulsating synthesiser effects. The singer added his own words as part of the finale:
They said we were too young
Our kind of love was no fun
But our love comes from above
Let's make ... love
Author Nicholas Pegg describes the recording as "faster and raunchier" than the Stones' performance with "a fresh, futuristic sheen",[22] while NME editors Roy Carr and Charles Shaar Murray considered Bowie to have performed "the unprecedented feat of beating the Stones on one of their own songs", remarking on the track's "polymorphous perversity" and "furious, coked-up drive".[23] However, Rolling Stone's contemporary review found the Bowie version "campy, butch, brittle and unsatisfying", suggesting that "one of the most ostensibly heterosexual calls in rock is made into a bi-anthem".[24]
Track listing[edit]
"Let's Spend the Night Together" (Mick Jagger, Keith Richards) – 3:03
"Lady Grinning Soul" (David Bowie) – 3:46
The Italian release featured "Watch That Man" on the B-side.
Personnel[edit]
David Bowie – guitar, harmonica, vocals
Mick Ronson – guitar, vocals
Trevor Bolder – bass
Mick "Woody" Woodmansey – drums
Mike Garson – piano, synthesizers
David Sanborn– saxophone
Live versions[edit]
A live version recorded by Bowie at the Hammersmith Odeon, London, on 3 July 1973 appears on the album Ziggy Stardust – The Motion Picture (1983).
Other releases[edit]
In addition to its appearance on Aladdin Sane, Bowie's version of "Let's Spend the Night Together" was included on the following compilations:
The Best of David Bowie (Japan 1974)
The Best of 1969/1974 (1997)
See also[edit]
List of songs banned by the BBC
References[edit]
Jump up ^ Gilliland, John (1969). "Show 46 – Sergeant Pepper at the Summit: The very best of a very good year. [Part 2]" (audio). Pop Chronicles. University of North Texas Libraries.
^ Jump up to: a b Let's Spend the Night Together at AllMusic
^ Jump up to: a b c Eder, Bruce (1989). Singles Collection: The London Years (Boxed set booklet). The Rolling Stones. New York City: ABKCO Records. p. 71. 1218-2.
Jump up ^ Wilmer, Valerie (December 1968). "Recording the Stones". Hit Parader. Magna Publishing Group: 9–11. Retrieved 16 July 2012.
Jump up ^ Billboard (21 January 1967). "Hot 100". Billboard. 79 (3): 25. ISSN 0006-2510.
Jump up ^ Billboard (4 March 1967). "Hot 100". Billboard. 79 (9): 24. ISSN 0006-2510.
Jump up ^ Whitburn, Joel (2015). The Comparison Book. Menonomee Falls, Wisconsin: Record Research Inc. p. 299. ISBN 978-0-89820-213-7.
Jump up ^ Christopher Sandford (1993, 1999). Mick Jagger: Primitive Cool: p.97
^ Jump up to: a b Dick Clark's 25 Years of Rock and Roll. 1981.
Jump up ^ "6 Ed Sullivan Shows Starring The Rolling Stones". Amazon.com. Retrieved 30 May 2012.
Jump up ^ "Stones tracks censored in China". BBC News. 7 April 2006. Retrieved 19 January 2008.
Jump up ^ "Official Website". The Rolling Stones. Retrieved 2014-05-24.
Jump up ^ "Austriancharts.at – The Rolling Stones – Let's Spend the Night Together" (in German). Ö3 Austria Top 40. Retrieved 17 June 2016.
Jump up ^ "Ultratop.be – The Rolling Stones – Let's Spend the Night Together" (in Dutch). Ultratop 50. Retrieved 17 June 2016.
Jump up ^ "Offiziellecharts.de – The Rolling Stones – Let's Spend the Night Together". GfK Entertainment Charts. Retrieved 17 June 2016.
Jump up ^ "The Irish Charts – Search Results – Let's Spend the Night Together". Irish Singles Chart. Retrieved 18 June 2016.
Jump up ^ "Norwegiancharts.com – The Rolling Stones – Let's Spend the Night Together". VG-lista. Retrieved 17 June 2016.
Jump up ^ "Rolling Stones: Artist Chart History". Official Charts Company. Retrieved 17 June 2016.
Jump up ^ "The Rolling Stones Chart History (Hot 100)". Billboard. Retrieved 17 June 2016.
Jump up ^ Cash Box Top 100 Singles, February 18, 1967
Jump up ^ https://dutchcharts.nl/showitem.asp?interpret=David+Bowie&titel=Let%27s+Spend+The+Night+Together&cat=s
Jump up ^ Nicholas Pegg (2000). The Complete David Bowie: pp.124-125
Jump up ^ Roy Carr & Charles Shaar Murray (1981). Bowie: An Illustrated Record: p.55
Jump up ^ Gerson, Ben (19 July 1973). "Aladdin Sane". Rolling Stone. Retrieved 19 January 2008.
External links[edit]
Lyrics of this song at MetroLyrics
show
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e
The Rolling Stones
Mick Jagger
Keith Richards
Charlie Watts
Ronnie Wood
Brian Jones
Ian Stewart
Mick Taylor
Bill Wyman
Tony Chapman
Ricky Fenson
Colin Golding
Carlo Little
Dick Taylor
Video releases
Let's Spend the Night Together (1983)
Stones at the Max (1992)
The Rolling Stones: Voodoo Lounge Live (1995)
The Rolling Stones Rock and Roll Circus (1996)
Bridges to Babylon Tour '97–98 (1998)
Four Flicks (2003)
The Biggest Bang (2007)
Ladies and Gentlemen: The Rolling Stones (2010)
Some Girls: Live in Texas '78 (2011)
Sweet Summer Sun: Live in Hyde Park (2013)
The Rolling Stones: Havana Moon (2016)
Documentaries
Charlie Is My Darling (1966)
The Stones in the Park (1969)
Gimme Shelter (1970)
Cocksucker Blues (1972)
Video Rewind (1984)
25×5: the Continuing Adventures of the Rolling Stones (1989)
Shine a Light (2008)
Stones in Exile (2010)
Crossfire Hurricane (2012)
Olé Olé Olé!: A Trip Across Latin America (2017)
Tours
British Tour 1963
1964 tours
1965 tours
1966 tours
European Tour 1967
American Tour 1969
European Tour 1970
UK Tour 1971
American Tour 1972
Pacific Tour 1973
European Tour 1973
Tour of the Americas '75
Tour of Europe '76
US Tour 1978
American Tour 1981
European Tour 1982
Steel Wheels/Urban Jungle Tour
Voodoo Lounge Tour
Bridges to Babylon Tour
No Security Tour
Licks Tour
A Bigger Bang
50 & Counting
14 On Fire
Zip Code Tour
América Latina Olé Tour 2016
No Filter Tour
Associated places
Redlands
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The Wick
Downe House
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Related
Discography
Songs
The Rolling Stones concerts
Jagger/Richards
Nanker Phelge
Rolling Stones Records
Promotone
Altamont Free Concert
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The Rolling Stones: An Illustrated Record
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Decca (UK) and
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singles
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"Not Fade Away" / "Little by Little"
"It's All Over Now"
"Tell Me" / "I Just Want to Make Love to You"
"Time Is on My Side"
"Little Red Rooster"
"Heart of Stone"
"The Last Time" / "Play with Fire"
"(I Can't Get No) Satisfaction" / "The Spider and the Fly"
"Get Off of My Cloud" / "I'm Free"
"As Tears Go By"
"19th Nervous Breakdown" / "As Tears Go By"
"Paint It Black" / "Stupid Girl"
"Mother's Little Helper" / "Lady Jane"
"Have You Seen Your Mother, Baby, Standing in the Shadow?"
"Let's Spend the Night Together" / "Ruby Tuesday"
"We Love You" / "Dandelion"
"In Another Land"
"She's a Rainbow" / "2000 Light Years from Home"
"Jumpin' Jack Flash"
"Street Fighting Man" / "No Expectations"
"Honky Tonk Women" / "You Can't Always Get What You Want"
Rolling Stones/
Atlantic singles
"Brown Sugar" / "Bitch" / "Let It Rock"
"Wild Horses" / "Sway"
"Tumbling Dice" / "Sweet Black Angel"
"Happy" / "All Down the Line"
"Angie" / "Silver Train"
"Doo Doo Doo Doo Doo (Heartbreaker)" / "Dancing with Mr. D"
"It's Only Rock 'n Roll (But I Like It)"
"Ain't Too Proud to Beg" / "Dance Little Sister"
"Fool to Cry"
"Hot Stuff"
Rolling Stones
"Miss You" / "Far Away Eyes"
"Beast of Burden" / "When the Whip Comes Down"
"Respectable" / "When the Whip Comes Down"
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Process capability index
From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
"Cpk" redirects here. For other uses, see CPK.
This article provides insufficient context for those unfamiliar with the subject. Please help improve the article with a good introductory style. (May 2015) (Learn how and when to remove this template message)
In process improvement efforts, the process capability index or process capability ratio is a statistical measure of process capability: the ability of a process to produce output within specification limits.[1] The concept of process capability only holds meaning for processes that are in a state of statistical control. Process capability indices measure how much "natural variation" a process experiences relative to its specification limits and allows different processes to be compared with respect to how well an organization controls them.
If the upper and lower specification limits of the process are USL and LSL, the target process mean is T, the estimated mean of the process is μ ^ {\displaystyle {\hat {\mu }}} and the estimated variability of the process (expressed as a standard deviation) is σ ^ {\displaystyle {\hat {\sigma }}} , then commonly accepted process capability indices include:
Index
Description
C ^ p = U S L − L S L 6 σ ^ {\displaystyle {\hat {C}}_{p}={\frac {USL-LSL}{6{\hat {\sigma }}}}} Estimates what the process is capable of producing if the process mean were to be centered between the specification limits. Assumes process output is approximately normally distributed.
C ^ p , l o w e r = μ ^ − L S L 3 σ ^ {\displaystyle {\hat {C}}_{p,lower}={{\hat {\mu }}-LSL \over 3{\hat {\sigma }}}} Estimates process capability for specifications that consist of a lower limit only (for example, strength). Assumes process output is approximately normally distributed.
C ^ p , u p p e r = U S L − μ ^ 3 σ ^ {\displaystyle {\hat {C}}_{p,upper}={USL-{\hat {\mu }} \over 3{\hat {\sigma }}}} Estimates process capability for specifications that consist of an upper limit only (for example, concentration). Assumes process output is approximately normally distributed.
C ^ p k = min [ U S L − μ ^ 3 σ ^ , μ ^ − L S L 3 σ ^ ] {\displaystyle {\hat {C}}_{pk}=\min {\Bigg [}{USL-{\hat {\mu }} \over 3{\hat {\sigma }}},{{\hat {\mu }}-LSL \over 3{\hat {\sigma }}}{\Bigg ]}} Estimates what the process is capable of producing, considering that the process mean may not be centered between the specification limits. (If the process mean is not centered, C ^ p {\displaystyle {\hat {C}}_{p}} overestimates process capability.) C ^ p k < 0 {\displaystyle {\hat {C}}_{pk}<0} if the process mean falls outside of the specification limits. Assumes process output is approximately normally distributed.
C ^ p m = C ^ p 1 + ( μ ^ − T σ ^ ) 2 {\displaystyle {\hat {C}}_{pm}={\frac {{\hat {C}}_{p}}{\sqrt {1+\left({\frac {{\hat {\mu }}-T}{\hat {\sigma }}}\right)^{2}}}}} Estimates process capability around a target, T. C ^ p m {\displaystyle {\hat {C}}_{pm}} is always greater than zero. Assumes process output is approximately normally distributed. C ^ p m {\displaystyle {\hat {C}}_{pm}} is also known as the Taguchi capability index.[2]
C ^ p k m = C ^ p k 1 + ( μ ^ − T σ ^ ) 2 {\displaystyle {\hat {C}}_{pkm}={\frac {{\hat {C}}_{pk}}{\sqrt {1+\left({\frac {{\hat {\mu }}-T}{\hat {\sigma }}}\right)^{2}}}}} Estimates process capability around a target, T, and accounts for an off-center process mean. Assumes process output is approximately normally distributed.
σ ^ {\displaystyle {\hat {\sigma }}} is estimated using the sample standard deviation.
Contents
1 Recommended values
2 Relationship to measures of process fallout
3 Example
4 See also
5 References
Recommended values[edit]
Process capability indices are constructed to express more desirable capability with increasingly higher values. Values near or below zero indicate processes operating off target ( μ ^ {\displaystyle {\hat {\mu }}} far from T) or with high variation.
Fixing values for minimum "acceptable" process capability targets is a matter of personal opinion, and what consensus exists varies by industry, facility, and the process under consideration. For example, in the automotive industry, the Automotive Industry Action Group sets forth guidelines in the Production Part Approval Process, 4th edition for recommended Cpk minimum values for critical-to-quality process characteristics. However, these criteria are debatable and several processes may not be evaluated for capability just because they have not properly been assessed.
Since the process capability is a function of the specification, the Process Capability Index is only as good as the specification. For instance, if the specification came from an engineering guideline without considering the function and criticality of the part, a discussion around process capability is useless, and would have more benefits if focused on what are the real risks of having a part borderline out of specification. The loss function of Taguchi better illustrates this concept.
At least one academic expert recommends[3] the following:
Situation
Recommended minimum process capability for two-sided specifications
Recommended minimum process capability for one-sided specification
Existing process 1.33 1.25
New process 1.50 1.45
Safety or critical parameter for existing process 1.50 1.45
Safety or critical parameter for new process 1.67 1.60
Six Sigma quality process 2.00 2.00
However where a process produces a characteristic with a capability index greater than 2.5, the unnecessary precision may be expensive.[4]
Relationship to measures of process fallout[edit]
The mapping from process capability indices, such as Cpk, to measures of process fallout is straightforward. Process fallout quantifies how many defects a process produces and is measured by DPMO or PPM. Process yield is the complement of process fallout and is approximately equal to the area under the probability density function Φ ( σ ) = 1 2 π ∫ − σ σ e − t 2 / 2 d t {\displaystyle \Phi (\sigma )={\frac {1}{\sqrt {2\pi }}}\int _{-\sigma }^{\sigma }e^{-t^{2}/2}\,dt} if the process output is approximately normally distributed.
In the short term ("short sigma"), the relationships are:
Cpk
Sigma level (σ)
Area under the
probability density function Φ ( σ ) {\displaystyle \Phi (\sigma )}
Process yield
Process fallout
(in terms of DPMO/PPM)
0.33 1 0.6826894921 68.27% 317311
0.67 2 0.9544997361 95.45% 45500
1.00 3 0.9973002039 99.73% 2700
1.33 4 0.9999366575 99.99% 63
1.67 5 0.9999994267 99.9999% 1
2.00 6 0.9999999980 99.9999998% 0.002
In the long term, processes can shift or drift significantly (most control charts are only sensitive to changes of 1.5σ or greater in process output). If there was a 1.5 sigma shift 1.5σ off of target in the processes (see Six Sigma), it would then produce these relationships:[5]
Cpk
Adjusted
Sigma level (σ)
Area under the
probability density function Φ ( σ ) {\displaystyle \Phi (\sigma )}
Process yield
Process fallout
(in terms of DPMO/PPM)
0.33 1 0.3085375387 30.85% 691462
0.67 2 0.6914624613 69.15% 308538
1.00 3 0.9331927987 93.32% 66807
1.33 4 0.9937903347 99.38% 6209
1.67 5 0.9997673709 99.9767% 232.6
2.00 6 0.9999966023 99.99966% 3.40
Because processes can shift or drift significantly long term, each process would have a unique sigma shift value, thus process capability indices are less applicable as they require statistical control.
Example[edit]
Consider a quality characteristic with target of 100.00 μm and upper and lower specification limits of 106.00 μm and 94.00 μm respectively. If, after carefully monitoring the process for a while, it appears that the process is in control and producing output predictably (as depicted in the run chart below), we can meaningfully estimate its mean and standard deviation.
If μ ^ {\displaystyle {\hat {\mu }}} and σ ^ {\displaystyle {\hat {\sigma }}} are estimated to be 98.94 μm and 1.03 μm, respectively, then
Index
C ^ p = U S L − L S L 6 σ ^ = 106.00 − 94.00 6 × 1.03 = 1.94 {\displaystyle {\hat {C}}_{p}={\frac {USL-LSL}{6{\hat {\sigma }}}}={\frac {106.00-94.00}{6\times 1.03}}=1.94}
C ^ p k = min [ U S L − μ ^ 3 σ ^ , μ ^ − L S L 3 σ ^ ] = min [ 106.00 − 98.94 3 × 1.03 , 98.94 − 94 3 × 1.03 ] = 1.60 {\displaystyle {\hat {C}}_{pk}=\min {\Bigg [}{USL-{\hat {\mu }} \over 3{\hat {\sigma }}},{{\hat {\mu }}-LSL \over 3{\hat {\sigma }}}{\Bigg ]}=\min {\Bigg [}{106.00-98.94 \over 3\times 1.03},{98.94-94 \over 3\times 1.03}{\Bigg ]}=1.60}
C ^ p m = C ^ p 1 + ( μ ^ − T σ ^ ) 2 = 1.94 1 + ( 98.94 − 100.00 1.03 ) 2 = 1.35 {\displaystyle {\hat {C}}_{pm}={\frac {{\hat {C}}_{p}}{\sqrt {1+\left({\frac {{\hat {\mu }}-T}{\hat {\sigma }}}\right)^{2}}}}={\frac {1.94}{\sqrt {1+\left({\frac {98.94-100.00}{1.03}}\right)^{2}}}}=1.35}
C ^ p k m = C ^ p k 1 + ( μ ^ − T σ ^ ) 2 = 1.60 1 + ( 98.94 − 100.00 1.03 ) 2 = 1.11 {\displaystyle {\hat {C}}_{pkm}={\frac {{\hat {C}}_{pk}}{\sqrt {1+\left({\frac {{\hat {\mu }}-T}{\hat {\sigma }}}\right)^{2}}}}={\frac {1.60}{\sqrt {1+\left({\frac {98.94-100.00}{1.03}}\right)^{2}}}}=1.11}
The fact that the process is running off-center (about 1σ below its target) is reflected in the markedly different values for Cp, Cpk, Cpm, and Cpkm.
See also[edit]
Process (engineering)
Process capability
Process performance index
References[edit]
Jump up ^ "What is Process Capability?". NIST/Sematech Engineering Statistics Handbook. National Institute of Standards and Technology. Retrieved 2008-06-22. External link in |work= (help)
Jump up ^ Boyles, Russell (1991). "The Taguchi Capability Index". Journal of Quality Technology. 23 (1). Milwaukee, Wisconsin: American Society for Quality Control. pp. 17–26. ISSN 0022-4065. OCLC 1800135.
Jump up ^ Montgomery, Douglas (2004). Introduction to Statistical Quality Control. New York, New York: John Wiley & Sons, Inc. p. 776. ISBN 978-0-471-65631-9. OCLC 56729567. Archived from the original on 2008-06-20.
Jump up ^ Booker, J. M.; Raines, M.; Swift, K. G. (2001). Designing Capable and Reliable Products. Oxford: Butterworth-Heinemann. ISBN 978-0-7506-5076-2. OCLC 47030836.
Jump up ^ "Sigma Conversion Calculator | BMGI.org". bmgi.org. Retrieved 2016-03-17.
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