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57-402: Speakon connectors are rated for at least 30 A RMS continuous current, higher than 1/4-inch TS phone connectors , two-pole twist lock, and XLR connectors for loudspeakers. A Speakon connector is designed with a locking system that may be designed for soldered or screw-type connections. Line connectors (usually FEMALE with the latch) mate with (usually MALE) panel connectors and typically

114-459: A continuous-time waveform ) is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous waveform. In the case of a set of n values { x 1 , x 2 , … , x n } {\displaystyle \{x_{1},x_{2},\dots ,x_{n}\}} , the RMS is The corresponding formula for

171-421: A population or a waveform x {\displaystyle x} is the RMS deviation of x {\displaystyle x} from its arithmetic mean x ¯ {\displaystyle {\bar {x}}} . They are related to the RMS value of x {\displaystyle x} by From this it is clear that the RMS value is always greater than or equal to

228-563: A sequence of positive weights w i we define the weighted power mean as M p ( x 1 , … , x n ) = ( ∑ i = 1 n w i x i p ∑ i = 1 n w i ) 1 / p {\displaystyle M_{p}(x_{1},\dots ,x_{n})=\left({\frac {\sum _{i=1}^{n}w_{i}x_{i}^{p}}{\sum _{i=1}^{n}w_{i}}}\right)^{{1}/{p}}} and when p = 0 , it

285-465: A 'combiner'. Also available are 2-pole "combo" receptacles that can also accept 4-pole cables and 1/4″ phone plugs. In 2019 the manufacturer introduced the STX series which included male line connectors and female panel connectors. Root mean square In mathematics , the root mean square (abbrev. RMS , RMS or rms ) of a set of numbers is the square root of the set's mean square . Given

342-457: A cable will have identical connectors at both ends. If it is needed to join cables, a coupler can be used (which essentially consists of two panel connectors mounted on the ends of a plastic tube). The design was conceived in 1987. Note that MANY users and even connector suppliers are confused by which is a male connector and which is a female, as the Female cable connector does look like it plugs INTO

399-426: A continuous function (or waveform) f ( t ) defined over the interval T 1 ≤ t ≤ T 2 {\displaystyle T_{1}\leq t\leq T_{2}} is and the RMS for a function over all time is The RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a continuous function or signal can be approximated by taking

456-425: A high-current speaker application. Speakon cables are intended solely for use in high current audio applications. Speakon connectors arrange their contacts in two concentric rings, with the inner contacts named +1, +2, etc. and the outer contacts connectors (in the four-pole and eight-pole version only) named −1, −2, etc. The phase convention is that positive voltage on the + contact causes air to be pushed away from

513-404: A pair of loudspeakers using a 'combiner' Y-lead connected to the two output channels, and a 'splitter' Y-lead to feed the loudspeakers. The 'combiner' and 'splitter' Y-leads are the same: two two-pole connectors on one end, connected to the ±1 and ±2 pins, respectively, of a four-pole line connector at the other end. Some amplifiers and mixer-amplifiers are configured to do this without the need for

570-492: A sequence of positive real numbers, then the following properties hold: In general, if p < q , then M p ( x 1 , … , x n ) ≤ M q ( x 1 , … , x n ) {\displaystyle M_{p}(x_{1},\dots ,x_{n})\leq M_{q}(x_{1},\dots ,x_{n})} and the two means are equal if and only if x 1 = x 2 = ... = x n . The inequality

627-402: A set x i {\displaystyle x_{i}} , its RMS is denoted as either x R M S {\displaystyle x_{\mathrm {RMS} }} or R M S x {\displaystyle \mathrm {RMS} _{x}} . The RMS is also known as the quadratic mean (denoted M 2 {\displaystyle M_{2}} ),

SECTION 10

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684-462: A signal. Standard deviation being the RMS of a signal's variation about the mean, rather than about 0, the DC component is removed (that is, RMS(signal) = stdev(signal) if the mean signal is 0). Generalized mean#Quadratic In mathematics , generalized means (or power mean or Hölder mean from Otto Hölder ) are a family of functions for aggregating sets of numbers. These include as special cases

741-418: A special case of the generalized mean . The RMS of a continuous function is denoted f R M S {\displaystyle f_{\mathrm {RMS} }} and can be defined in terms of an integral of the square of the function. In estimation theory , the root-mean-square deviation of an estimator measures how far the estimator strays from the data. The RMS value of a set of values (or

798-558: Is as follows: Define the following function: f  : R + → R + f ( x ) = x q p {\displaystyle f(x)=x^{\frac {q}{p}}} . f is a power function, so it does have a second derivative: f ″ ( x ) = ( q p ) ( q p − 1 ) x q p − 2 {\displaystyle f''(x)=\left({\frac {q}{p}}\right)\left({\frac {q}{p}}-1\right)x^{{\frac {q}{p}}-2}} which

855-612: Is equal to the weighted geometric mean : M 0 ( x 1 , … , x n ) = ( ∏ i = 1 n x i w i ) 1 / ∑ i = 1 n w i . {\displaystyle M_{0}(x_{1},\dots ,x_{n})=\left(\prod _{i=1}^{n}x_{i}^{w_{i}}\right)^{1/\sum _{i=1}^{n}w_{i}}.} The unweighted means correspond to setting all w i = 1/n . A few particular values of p yield special cases with their own names: For

912-678: Is negative, and q is positive, the inequality is equivalent to the one proved above: ( ∑ i = 1 n w i x i p ) 1 / p ≤ ∏ i = 1 n x i w i ≤ ( ∑ i = 1 n w i x i q ) 1 / q {\displaystyle \left(\sum _{i=1}^{n}w_{i}x_{i}^{p}\right)^{1/p}\leq \prod _{i=1}^{n}x_{i}^{w_{i}}\leq \left(\sum _{i=1}^{n}w_{i}x_{i}^{q}\right)^{1/q}} The proof for positive p and q

969-420: Is not true for an arbitrary waveform, which may not be periodic or continuous. For a zero-mean sine wave, the relationship between RMS and peak-to-peak amplitude is: For other waveforms, the relationships are not the same as they are for sine waves. For example, for either a triangular or sawtooth wave: Waveforms made by summing known simple waveforms have an RMS value that is the root of the sum of squares of

1026-403: Is periodic (such as household AC power), it is still meaningful to discuss the average power dissipated over time, which is calculated by taking the average power dissipation: So, the RMS value, I RMS , of the function I ( t ) is the constant current that yields the same power dissipation as the time-averaged power dissipation of the current I ( t ). Average power can also be found using

1083-977: Is strictly positive within the domain of f , since q > p , so we know f is convex. Using this, and the Jensen's inequality we get: f ( ∑ i = 1 n w i x i p ) ≤ ∑ i = 1 n w i f ( x i p ) ( ∑ i = 1 n w i x i p ) q / p ≤ ∑ i = 1 n w i x i q {\displaystyle {\begin{aligned}f\left(\sum _{i=1}^{n}w_{i}x_{i}^{p}\right)&\leq \sum _{i=1}^{n}w_{i}f(x_{i}^{p})\\[3pt]\left(\sum _{i=1}^{n}w_{i}x_{i}^{p}\right)^{q/p}&\leq \sum _{i=1}^{n}w_{i}x_{i}^{q}\end{aligned}}} after raising both side to

1140-585: Is the sampling period, where X [ m ] = DFT ⁡ { x [ n ] } {\displaystyle X[m]=\operatorname {DFT} \{x[n]\}} and N is the sample size, that is, the number of observations in the sample and DFT coefficients. In this case, the RMS computed in the time domain is the same as in the frequency domain: The standard deviation σ x = ( x − x ¯ ) rms {\displaystyle \sigma _{x}=(x-{\overline {x}})_{\text{rms}}} of

1197-480: Is true for real values of p and q , as well as positive and negative infinity values. It follows from the fact that, for all real p , ∂ ∂ p M p ( x 1 , … , x n ) ≥ 0 {\displaystyle {\frac {\partial }{\partial p}}M_{p}(x_{1},\dots ,x_{n})\geq 0} which can be proved using Jensen's inequality . In particular, for p in {−1, 0, 1} ,

SECTION 20

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1254-836: The Pythagorean means ( arithmetic , geometric , and harmonic means ). If p is a non-zero real number , and x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} are positive real numbers, then the generalized mean or power mean with exponent p of these positive real numbers is M p ( x 1 , … , x n ) = ( 1 n ∑ i = 1 n x i p ) 1 / p . {\displaystyle M_{p}(x_{1},\dots ,x_{n})=\left({\frac {1}{n}}\sum _{i=1}^{n}x_{i}^{p}\right)^{{1}/{p}}.} (See p -norm ). For p = 0 we set it equal to

1311-450: The exponential function to both sides and observing that as a strictly increasing function it preserves the sign of the inequality, we get ∏ i = 1 n x i w i ≤ ∑ i = 1 n w i x i . {\displaystyle \prod _{i=1}^{n}x_{i}^{w_{i}}\leq \sum _{i=1}^{n}w_{i}x_{i}.} Taking q -th powers of

1368-409: The power , P , dissipated by an electrical resistance , R . It is easy to do the calculation when there is a constant current , I , through the resistance. For a load of R ohms, power is given by: However, if the current is a time-varying function, I ( t ), this formula must be extended to reflect the fact that the current (and thus the instantaneous power) is varying over time. If the function

1425-763: The x i yields ∏ i = 1 n x i q ⋅ w i ≤ ∑ i = 1 n w i x i q ∏ i = 1 n x i w i ≤ ( ∑ i = 1 n w i x i q ) 1 / q . {\displaystyle {\begin{aligned}&\prod _{i=1}^{n}x_{i}^{q{\cdot }w_{i}}\leq \sum _{i=1}^{n}w_{i}x_{i}^{q}\\&\prod _{i=1}^{n}x_{i}^{w_{i}}\leq \left(\sum _{i=1}^{n}w_{i}x_{i}^{q}\right)^{1/q}.\end{aligned}}} Thus, we are done for

1482-479: The RMS of a sample consisting of equally spaced observations. Additionally, the RMS value of various waveforms can also be determined without calculus , as shown by Cartwright. In the case of the RMS statistic of a random process , the expected value is used instead of the mean. If the waveform is a pure sine wave , the relationships between amplitudes (peak-to-peak, peak) and RMS are fixed and known, as they are for any continuous periodic wave. However, this

1539-496: The above formula, which implies V P  =  V RMS  ×  √ 2 , assuming the source is a pure sine wave. Thus the peak value of the mains voltage in the USA is about 120 ×  √ 2 , or about 170 volts. The peak-to-peak voltage, being double this, is about 340 volts. A similar calculation indicates that the peak mains voltage in Europe is about 325 volts, and

1596-506: The analogous equation for sinusoidal voltage: where I P represents the peak current and V P represents the peak voltage. Because of their usefulness in carrying out power calculations, listed voltages for power outlets (for example, 120   V in the US, or 230   V in Europe) are almost always quoted in RMS values, and not peak values. Peak values can be calculated from RMS values from

1653-417: The average, in that the RMS includes the squared deviation (error) as well. Physical scientists often use the term root mean square as a synonym for standard deviation when it can be assumed the input signal has zero mean, that is, referring to the square root of the mean squared deviation of a signal from a given baseline or fit. This is useful for electrical engineers in calculating the "AC only" RMS of

1710-454: The component RMS values, if the component waveforms are orthogonal (that is, if the average of the product of one simple waveform with another is zero for all pairs other than a waveform times itself). Alternatively, for waveforms that are perfectly positively correlated, or "in phase" with each other, their RMS values sum directly. The RMS of an alternating electric current equals the value of constant direct current that would dissipate

1767-2766: The continuity of the exponential function, we can substitute back into the above relation to obtain lim p → 0 M p ( x 1 , … , x n ) = exp ⁡ ( ln ⁡ ( ∏ i = 1 n x i w i ) ) = ∏ i = 1 n x i w i = M 0 ( x 1 , … , x n ) {\displaystyle \lim _{p\to 0}M_{p}(x_{1},\dots ,x_{n})=\exp {\left(\ln {\left(\prod _{i=1}^{n}x_{i}^{w_{i}}\right)}\right)}=\prod _{i=1}^{n}x_{i}^{w_{i}}=M_{0}(x_{1},\dots ,x_{n})} as desired. Assume (possibly after relabeling and combining terms together) that x 1 ≥ ⋯ ≥ x n {\displaystyle x_{1}\geq \dots \geq x_{n}} . Then lim p → ∞ M p ( x 1 , … , x n ) = lim p → ∞ ( ∑ i = 1 n w i x i p ) 1 / p = x 1 lim p → ∞ ( ∑ i = 1 n w i ( x i x 1 ) p ) 1 / p = x 1 = M ∞ ( x 1 , … , x n ) . {\displaystyle {\begin{aligned}\lim _{p\to \infty }M_{p}(x_{1},\dots ,x_{n})&=\lim _{p\to \infty }\left(\sum _{i=1}^{n}w_{i}x_{i}^{p}\right)^{1/p}\\&=x_{1}\lim _{p\to \infty }\left(\sum _{i=1}^{n}w_{i}\left({\frac {x_{i}}{x_{1}}}\right)^{p}\right)^{1/p}\\&=x_{1}=M_{\infty }(x_{1},\dots ,x_{n}).\end{aligned}}} The formula for M − ∞ {\displaystyle M_{-\infty }} follows from M − ∞ ( x 1 , … , x n ) = 1 M ∞ ( 1 / x 1 , … , 1 / x n ) = x n . {\displaystyle M_{-\infty }(x_{1},\dots ,x_{n})={\frac {1}{M_{\infty }(1/x_{1},\dots ,1/x_{n})}}=x_{n}.} Let x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} be

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1824-766: The exponential function as M p ( x 1 , … , x n ) = exp ⁡ ( ln ⁡ [ ( ∑ i = 1 n w i x i p ) 1 / p ] ) = exp ⁡ ( ln ⁡ ( ∑ i = 1 n w i x i p ) p ) {\displaystyle M_{p}(x_{1},\dots ,x_{n})=\exp {\left(\ln {\left[\left(\sum _{i=1}^{n}w_{i}x_{i}^{p}\right)^{1/p}\right]}\right)}=\exp {\left({\frac {\ln {\left(\sum _{i=1}^{n}w_{i}x_{i}^{p}\right)}}{p}}\right)}} In

1881-520: The fact the logarithm is concave: log ⁡ ∏ i = 1 n x i w i = ∑ i = 1 n w i log ⁡ x i ≤ log ⁡ ∑ i = 1 n w i x i . {\displaystyle \log \prod _{i=1}^{n}x_{i}^{w_{i}}=\sum _{i=1}^{n}w_{i}\log x_{i}\leq \log \sum _{i=1}^{n}w_{i}x_{i}.} By applying

1938-460: The following inequality holds: ( ∑ i = 1 n w i x i p ) 1 / p ≤ ( ∑ i = 1 n w i x i q ) 1 / q {\displaystyle \left(\sum _{i=1}^{n}w_{i}x_{i}^{p}\right)^{1/p}\leq \left(\sum _{i=1}^{n}w_{i}x_{i}^{q}\right)^{1/q}} if p

1995-432: The four connections for the higher-frequency signal, with the other two for the lower-frequency signal) without two separate cables. Similarly, the eight-pole connector could be used for tri-amping (two poles each for low, mid and high frequencies with two unused), or quad-amping (two poles each for high, mid, low and sub). Another use for the four-pole cable is to carry two channels of amplified signal from an amplifier to

2052-1640: The generalized mean inequality implies the Pythagorean means inequality as well as the inequality of arithmetic and geometric means . We will prove the weighted power mean inequality. For the purpose of the proof we will assume the following without loss of generality: w i ∈ [ 0 , 1 ] ∑ i = 1 n w i = 1 {\displaystyle {\begin{aligned}w_{i}\in [0,1]\\\sum _{i=1}^{n}w_{i}=1\end{aligned}}} The proof for unweighted power means can be easily obtained by substituting w i = 1/ n . Suppose an average between power means with exponents p and q holds: ( ∑ i = 1 n w i x i p ) 1 / p ≥ ( ∑ i = 1 n w i x i q ) 1 / q {\displaystyle \left(\sum _{i=1}^{n}w_{i}x_{i}^{p}\right)^{1/p}\geq \left(\sum _{i=1}^{n}w_{i}x_{i}^{q}\right)^{1/q}} applying this, then: ( ∑ i = 1 n w i x i p ) 1 / p ≥ ( ∑ i = 1 n w i x i q ) 1 / q {\displaystyle \left(\sum _{i=1}^{n}{\frac {w_{i}}{x_{i}^{p}}}\right)^{1/p}\geq \left(\sum _{i=1}^{n}{\frac {w_{i}}{x_{i}^{q}}}\right)^{1/q}} We raise both sides to

2109-431: The geometric mean (which is the limit of means with exponents approaching zero, as proved below): M 0 ( x 1 , … , x n ) = ( ∏ i = 1 n x i ) 1 / n . {\displaystyle M_{0}(x_{1},\dots ,x_{n})=\left(\prod _{i=1}^{n}x_{i}\right)^{1/n}.} Furthermore, for

2166-437: The geometric mean without using a limit with f ( x ) = log( x ) . The power mean is obtained for f ( x ) = x . Properties of these means are studied in de Carvalho (2016). A power mean serves a non-linear moving average which is shifted towards small signal values for small p and emphasizes big signal values for big p . Given an efficient implementation of a moving arithmetic mean called smooth one can implement

2223-944: The inequality for means with exponents − p and − q , and we can use the same reasoning backwards, thus proving the inequalities to be equivalent, which will be used in some of the later proofs. For any q > 0 and non-negative weights summing to 1, the following inequality holds: ( ∑ i = 1 n w i x i − q ) − 1 / q ≤ ∏ i = 1 n x i w i ≤ ( ∑ i = 1 n w i x i q ) 1 / q . {\displaystyle \left(\sum _{i=1}^{n}w_{i}x_{i}^{-q}\right)^{-1/q}\leq \prod _{i=1}^{n}x_{i}^{w_{i}}\leq \left(\sum _{i=1}^{n}w_{i}x_{i}^{q}\right)^{1/q}.} The proof follows from Jensen's inequality , making use of

2280-504: The inequality with positive q ; the case for negatives is identical but for the swapped signs in the last step: ∏ i = 1 n x i − q ⋅ w i ≤ ∑ i = 1 n w i x i − q . {\displaystyle \prod _{i=1}^{n}x_{i}^{-q{\cdot }w_{i}}\leq \sum _{i=1}^{n}w_{i}x_{i}^{-q}.} Of course, taking each side to

2337-2153: The limit p → 0 , we can apply L'Hôpital's rule to the argument of the exponential function. We assume that p ∈ R {\displaystyle p\in \mathbb {R} } but p ≠ 0 , and that the sum of w i is equal to 1 (without loss in generality); Differentiating the numerator and denominator with respect to p , we have lim p → 0 ln ⁡ ( ∑ i = 1 n w i x i p ) p = lim p → 0 ∑ i = 1 n w i x i p ln ⁡ x i ∑ j = 1 n w j x j p 1 = lim p → 0 ∑ i = 1 n w i x i p ln ⁡ x i ∑ j = 1 n w j x j p = ∑ i = 1 n w i ln ⁡ x i ∑ j = 1 n w j = ∑ i = 1 n w i ln ⁡ x i = ln ⁡ ( ∏ i = 1 n x i w i ) {\displaystyle {\begin{aligned}\lim _{p\to 0}{\frac {\ln {\left(\sum _{i=1}^{n}w_{i}x_{i}^{p}\right)}}{p}}&=\lim _{p\to 0}{\frac {\frac {\sum _{i=1}^{n}w_{i}x_{i}^{p}\ln {x_{i}}}{\sum _{j=1}^{n}w_{j}x_{j}^{p}}}{1}}\\&=\lim _{p\to 0}{\frac {\sum _{i=1}^{n}w_{i}x_{i}^{p}\ln {x_{i}}}{\sum _{j=1}^{n}w_{j}x_{j}^{p}}}\\&={\frac {\sum _{i=1}^{n}w_{i}\ln {x_{i}}}{\sum _{j=1}^{n}w_{j}}}\\&=\sum _{i=1}^{n}w_{i}\ln {x_{i}}\\&=\ln {\left(\prod _{i=1}^{n}x_{i}^{w_{i}}\right)}\end{aligned}}} By

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2394-408: The load, R , is purely resistive). Reactive loads (that is, loads capable of not just dissipating energy but also storing it) are discussed under the topic of AC power . In the common case of alternating current when I ( t ) is a sinusoidal current, as is approximately true for mains power, the RMS value is easy to calculate from the continuous case equation above. If I p is defined to be

2451-406: The long term. The term RMS power is sometimes erroneously used (e.g., in the audio industry) as a synonym for mean power or average power (it is proportional to the square of the RMS voltage or RMS current in a resistive load). For a discussion of audio power measurements and their shortcomings, see Audio power . In the physics of gas molecules, the root-mean-square speed is defined as

2508-435: The male chassis connector. The easiest way to remember it is to look at Neutrik part numbers where an "F" indicates Female and an "M denotes Male (eg "NL4FX" is the 'normal' cable connector (with a latch). Speakon connectors are designed to be unambiguous in their use in speaker cables. With 1/4" speaker jacks and XLR connections, it is possible for users to erroneously use low-current shielded microphone or instrument cables in

2565-401: The order of thousands of km/h, even though the average velocity of its molecules is zero. When two data sets — one set from theoretical prediction and the other from actual measurement of some physical variable, for instance — are compared, the RMS of the pairwise differences of the two data sets can serve as a measure of how far on average the error is from 0. The mean of the absolute values of

2622-495: The pairwise differences could be a useful measure of the variability of the differences. However, the RMS of the differences is usually the preferred measure, probably due to mathematical convention and compatibility with other formulae. The RMS can be computed in the frequency domain, using Parseval's theorem . For a sampled signal x [ n ] = x ( t = n T ) {\displaystyle x[n]=x(t=nT)} , where T {\displaystyle T}

2679-459: The peak current, then: where t is time and ω is the angular frequency ( ω  = 2 π / T , where T is the period of the wave). Since I p is a positive constant and was to be squared within the integral: Using a trigonometric identity to eliminate squaring of trig function: but since the interval is a whole number of complete cycles (per definition of RMS), the sine terms will cancel out, leaving: A similar analysis leads to

2736-428: The peak-to-peak mains voltage, about 650 volts. RMS quantities such as electric current are usually calculated over one cycle. However, for some purposes the RMS current over a longer period is required when calculating transmission power losses. The same principle applies, and (for example) a current of 10 amps used for 12 hours each 24-hour day represents an average current of 5 amps, but an RMS current of 7.07 amps, in

2793-552: The power of 1/ q (an increasing function, since 1/ q is positive) we get the inequality which was to be proven: ( ∑ i = 1 n w i x i p ) 1 / p ≤ ( ∑ i = 1 n w i x i q ) 1 / q {\displaystyle \left(\sum _{i=1}^{n}w_{i}x_{i}^{p}\right)^{1/p}\leq \left(\sum _{i=1}^{n}w_{i}x_{i}^{q}\right)^{1/q}} Using

2850-513: The power of a negative number -1/ q swaps the direction of the inequality. ∏ i = 1 n x i w i ≥ ( ∑ i = 1 n w i x i − q ) − 1 / q . {\displaystyle \prod _{i=1}^{n}x_{i}^{w_{i}}\geq \left(\sum _{i=1}^{n}w_{i}x_{i}^{-q}\right)^{-1/q}.} We are to prove that for any p < q

2907-1035: The power of −1 (strictly decreasing function in positive reals): ( ∑ i = 1 n w i x i − p ) − 1 / p = ( 1 ∑ i = 1 n w i 1 x i p ) 1 / p ≤ ( 1 ∑ i = 1 n w i 1 x i q ) 1 / q = ( ∑ i = 1 n w i x i − q ) − 1 / q {\displaystyle \left(\sum _{i=1}^{n}w_{i}x_{i}^{-p}\right)^{-1/p}=\left({\frac {1}{\sum _{i=1}^{n}w_{i}{\frac {1}{x_{i}^{p}}}}}\right)^{1/p}\leq \left({\frac {1}{\sum _{i=1}^{n}w_{i}{\frac {1}{x_{i}^{q}}}}}\right)^{1/q}=\left(\sum _{i=1}^{n}w_{i}x_{i}^{-q}\right)^{-1/q}} We get

SECTION 50

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2964-628: The previously shown equivalence we can prove the inequality for negative p and q by replacing them with −q and −p , respectively. The power mean could be generalized further to the generalized f -mean : M f ( x 1 , … , x n ) = f − 1 ( 1 n ⋅ ∑ i = 1 n f ( x i ) ) {\displaystyle M_{f}(x_{1},\dots ,x_{n})=f^{-1}\left({{\frac {1}{n}}\cdot \sum _{i=1}^{n}{f(x_{i})}}\right)} This covers

3021-427: The purpose of the proof, we will assume without loss of generality that w i ∈ [ 0 , 1 ] {\displaystyle w_{i}\in [0,1]} and ∑ i = 1 n w i = 1. {\displaystyle \sum _{i=1}^{n}w_{i}=1.} We can rewrite the definition of M p {\displaystyle M_{p}} using

3078-462: The same method that in the case of a time-varying voltage , V ( t ), with RMS value V RMS , This equation can be used for any periodic waveform , such as a sinusoidal or sawtooth waveform , allowing us to calculate the mean power delivered into a specified load. By taking the square root of both these equations and multiplying them together, the power is found to be: Both derivations depend on voltage and current being proportional (that is,

3135-443: The same power in a resistive load . A special case of RMS of waveform combinations is: where V DC {\displaystyle {\text{V}}_{\text{DC}}} refers to the direct current (or average) component of the signal, and RMS AC {\displaystyle {\text{RMS}}_{\text{AC}}} is the alternating current component of the signal. Electrical engineers often need to know

3192-450: The speaker. Speakon connectors are made in two, four and eight-pole configurations. The two-pole line connector will mate with the four-pole panel connector, connecting to +1 and −1; but the reverse combination will not work. The eight-pole connector is physically larger to accommodate the extra poles. The four-pole connector is the most common at least from the availability of ready-made leads, as it allows for things like bi-amping (two of

3249-429: The square root of the average squared-speed. The RMS speed of an ideal gas is calculated using the following equation: where R represents the gas constant , 8.314 J/(mol·K), T is the temperature of the gas in kelvins , and M is the molar mass of the gas in kilograms per mole. In physics, speed is defined as the scalar magnitude of velocity. For a stationary gas, the average speed of its molecules can be in

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