RMS Value

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  Averages and Means (Concepts & Applications)

  • (Author)
  • 2014(Publication Date)
  • Library Press

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter 3 Fundamental Concepts of Averages and Means Root mean square In mathematics, the root mean square (abbreviated RMS or rms ), also known as the quadratic mean , is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids. RMS is used in various fields, including electrical engineering ; one of the more prominent uses of RMS is in the field of signal amplifiers. It can be calculated for a series of discrete values or for a continuously varying function. The name comes from the fact that it is the square root of the mean of the squares of the values. It is a special case of the generalized mean with the exponent p = 2. Definition The RMS Value of a set of values (or a continuous-time waveform) is the square root of the arithmetic mean (average) of the squares of the original values (or the square of the function that defines the continuous waveform). In the case of a set of n values , the RMS Value is given by: The corresponding formula for a continuous function (or waveform) f ( t ) defined over the interval is ________________________ WORLD TECHNOLOGIES ________________________ 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 the RMS of a series of equally spaced samples. 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.

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  Fundamental Concepts and Applications of Averages and Means

  • (Author)
  • 2014(Publication Date)
  • Library Press

    (Publisher)

  ________________________ WORLD TECHNOLOGIES ________________________ Chapter 4 Fundamental Concepts of Averages and Means Root mean square In mathematics, the root mean square (abbreviated RMS or rms ), also known as the quadratic mean , is a statistical measure of the magnitude of a varying quantity. It is especially useful when variates are positive and negative, e.g., sinusoids. RMS is used in various fields, including electrical engineering; one of the more prominent uses of RMS is in the field of signal amplifiers. It can be calculated for a series of discrete values or for a continuously varying function. The name comes from the fact that it is the square root of the mean of the squares of the values. It is a special case of the generalized mean with the exponent p = 2. Definition The RMS Value of a set of values (or a continuous-time waveform) is the square root of the arithmetic mean (average) of the squares of the original values (or the square of the function that defines the continuous waveform). In the case of a set of n values , the RMS Value is given by: The corresponding formula for a continuous function (or waveform) f ( t ) defined over the interval is ________________________ WORLD TECHNOLOGIES ________________________ 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 the RMS of a series of equally spaced samples. 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.

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  Electrical Engineering

  Fundamentals

  • Viktor Hacker, Christof Sumereder(Authors)
  • 2020(Publication Date)
  • De Gruyter Oldenbourg

    (Publisher)

  This leads to simplifications when using AC technology. The instantaneous values, in practice, are often not important as it does not matter e.g. with electric heating if the level of current, that causes the heating effect, alternates sinusoidally or if it is a direct current with the same heating effect (definition of the effective value!). For a load, the root mean square (=effective value) of power is significant. Figure 7.3: Effective value (RMS). Mathematical derivation: When comparing the power of an ohmic resistor passed through by direct current in one case and by alternating current in the other, the following considerations apply: In the case of direct current, the following applies for the power: P = V ⋅ I = I 2 ⋅ R = V 2 R Correspondingly, in the case of alternating current, the following applies: P (t) = i 2 (t) ⋅ R If the sinusoidally alternating current i t = i ˆ ⋅ s i n ω t is squared, this results. in: i 2 (t) = i ˆ 2 ⋅ sin 2 (ω t) The temporal sequences of the quantities are represented in Figure 7.4. Figure 7.4: Effective value. The mean value of the curve i 2 (t) is a direct quantity 82, it is designated as I 2. For the integral over the time period T, this results in: I 2 = 1 T ⋅ ∫ 0 T i 2 (t) ⋅ d t = i ˆ 2 T ⋅ ∫ 0 T s i n 2 (ω t) d t = i ˆ 2 ω T ∫ 0 T s i n 2 (ω t) ⋅ d (ω t) = i ˆ 2 2 For I this leads. to: I = 1 T ⋅ ∫ 0 T i 2 (t) ⋅ d t = i ˆ 2 2 = i ˆ 2 The resulting current value for I is called root mean square or effective value. Considering P = V 2 R, an analogue derivation 83 can be performed for voltage. We receive: V = v ˆ 2 If there is no additional indication regarding the specification of alternating voltage or alternating current, it is always the effective value that is intended. The alternating voltage of 230 V commonly used in households is also designated by its effective value

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  Electrical, Electronics, and Digital Hardware Essentials for Scientists and Engineers

  • Ed Lipiansky(Author)
  • 2012(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  pushed ” by the AC voltage source, is the cause of heat being produced in the immediate vicinity of the toaster-heating element. The outlet on the kitchen wall is the point where we connect the appliance to the AC voltage source. The AC volt­age source from the electric utility company is usually located in a remote site, far away from the home. In most households in the Unites States, the standard AC voltage is 120 V. The 120 V refers to the root mean square (RMS) value of the sinusoidal waveform, where RMS is defined mathematically by the following equation:

  (2.1)  

  Equation (2.1) is the RMS Value of waveform f (t ).

  In Equation (2.1) T is the period of the waveform. The waveform f (t ) can be either a voltage or a current, and t is the time, the independent variable. RMS of a waveform is also referred to as the effective value of the waveform.

  When f (t ) is a sinusoidal waveform such as v (t ) = V sin (ωt  + θ ); V is the amplitude (or peak value) of the sinusoidal waveform in volts, ω is its angular frequency equal to 2πf , where f equals the inverse of the sinusoid’s period T or the sinusoid frequency, given in units of second−1 or hertz, and θ is the sinusoidal waveform phase shift. The units of the angular frequency ω are given in radians per second. Solving Equation (2.1) for a sinusoidal voltage, the RMS Value of it is

  (2.2)  

  where V is the peak value or magnitude of the sinusoidal waveform. So the 120 V at the kitchen outlet is the RMS Value of the sinusoidal waveform that the electric utility company provides to U.S. households. Also applying Equation (2.2) to a current waveform, i (t ) = I sin (ωt  + θ ), we find that its RMS Value is also

  (2.3)  

  In Equation (2.3) I is the peak value or amplitude of the current waveform.

  2.1.1  Ideal and Real AC Voltage Sources

  An ideal AC voltage source is one that produces a sinusoidal voltage that varies with time. Most importantly, the amplitude and the RMS Value of such voltage source does not vary based on how much current the load across the source terminals is drawing. This means that the internal resistance of an ideal AC voltage source is zero. So whether the voltage source supplies no current or very large currents, the voltage amplitude and RMS Value remain constant. It is also true that the waveforms retain their sinusoidal shape and original frequency f and phase angle θ . On the other hand, a real AC voltage source amplitude does not remain constant with the level of current being supplied by the real AC voltage source. This concept is similar to that of ideal and real DC voltage sources. The real AC voltage source can be modeled as an ideal AC voltage source in series with its internal resistance, the real AC voltage source internal resistance is not zero, and it is a finite number as shown by Figure 2.1

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  The RF Transmission Systems Handbook

  • Jerry C. Whitaker(Author)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  20RF Voltage and Power Measurement Jerry C. Whitaker Editor 20.1 Introduction Root Mean Square Average-Response Measurement Peak-Response Measurement Measurement Bandwidth Meter Accuracy 20.2 RF Power Measurement Decibel Measurement Noise Measurement Phase Measurement Nonlinear Distortion Intermodulation Distortion Measurement Techniques Addition and Cancellation of Distortion Components

  20.1 Introduction

  The simplest definition of a level measurement is “the alternating current amplitude at a particular place in the system under test.” However, in contrast to direct current measurements, there are many ways of specifying ac voltage in a circuit. The most common methods include:

  • Average response
  • Root mean square (rms)
  • Peak response

  Strictly speaking, the term level refers to a logarithmic, or decibel, measurement. However, common parlance employs the term for an ac amplitude measurement, and that convention will be followed in this chapter.

  Root Mean Square

  The root-mean-square (rms) technique measures the effective power of the ac signal. It specifies the value of the dc equivalent that would dissipate the same power if either were applied to a load resistor. This process is illustrated in Fig. 20.1 for voltage measurements. The input signal is squared, and the average value is found. This is equivalent to finding the average power. The square root of this value is taken to transfer the signal from a power value back to a voltage. For the case of a sine wave, the RMS Value is 0.707 of its maximum value.

  Assume that the signal is no longer a sine wave but rather a sine wave and several harmonics. If the rms amplitude of each harmonic is measured individually and added, the resulting value will be the same as an rms measurement of the signals together. Because rms voltages cannot be added directly, it is necessary to perform an rms addition. Each voltage is squared, and the squared values are added as follows:

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  Basic Engineering Circuit Analysis

  • J. David Irwin, R. Mark Nelms(Authors)
  • 2021(Publication Date)
  • Wiley

    (Publisher)

  9.4 Effective or RMS Values 323 On using the RMS Values for voltage and current, the average power can be written, in general, as P = V rms I rms cos (θ υ − θ i ) 9.25 The power absorbed by a resistor R is P = I 2 rms R = V 2 rms _ R 9.26 In dealing with voltages and currents in numerous electrical applications, it is important to know whether the values quoted are maximum, average, rms, or what. We are familiar with the 120-V ac electrical outlets in our home. In this case, the 120 V is the RMS Value of the volt- age in our home. The maximum or peak value of this voltage is 120 √ — 2 = 170 V. The voltage at our electrical outlets could be written as 170 cos 377t V. The maximum or peak value must be given if we write the voltage in this form. There should be no question in our minds that this is the peak value. It is common practice to specify the voltage rating of ac electrical devices in terms of the rms voltage. For example, if you examine an incandescent light bulb, you will see a voltage rating of 120 V, which is the RMS Value. For now, we will add an rms to our voltages and currents to indicate that we are using RMS Values in our calculations. Solution Substituting these expressions into Eq. (9.23) yields I rms = [ 1 _ T  0 T I 2 M cos 2 (ωt − θ) dt ] 1/2 Using the trigonometric identity cos 2 ϕ = 1 _ 2 + 1 _ 2 cos 2ϕ we find that the preceding equation can be expressed as I rms = I M { ω ___ 2π  0 2π/ω [ 1 _ 2 + 1 _ 2 cos (2ωt − 2θ) ] dt } 1/2 Since we know that the average or mean value of a cosine wave is zero, I rms = I M ( ω ___ 2π  0 2π/ω 1 __ 2 dt ) 1/2 = I M [ ω _ 2π ( t _ 2 )  0 2π/ω ] 1/2 = I M _ √ — 2 9.24 Therefore, the RMS Value of a sinusoid is equal to the maximum value divided by √ — 2. Hence, a sinusoi- dal current with a maximum value of I M delivers the same average power to a resistor R as a dc current with a value of I M / √ — 2. Recall that earlier a phasor X was defined as X M / θ for a sinusoidal wave of the form X M cos (ωt + θ).

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  Practical Audio Electronics

  • Kevin Robinson(Author)
  • 2020(Publication Date)
  • Focal Press

    (Publisher)

  For a sine wave the peak-to-peak measurement is exactly twice the peak value. This is not necessarily the case for more complex signals, which may often be asymmetrical above and below the centreline. Strictly speaking, the average value of a sine wave is zero, since the waveform is symmetrical around the centre or zero line. What has been marked as ‘average’ on Figure 5.8 might more accurately be described as the half-wave average or rectified average, where the portion of the signal below the zero line is reflected up to lie above it (see later in this chapter for a discussion of signal rectification). Here it could most simply be characterised as the average signal level as measured between 0 ◦ and 180 ◦ (or indeed 0 ◦ and 90 ◦ , which would yield the same result due to the symmetry of the waveform). This rectified average level is a simple thing to measure for an electrical signal, requiring only a straightforward circuit in order to perform the task. Unfortu-nately, its ease of measurement notwithstanding, it is a far less useful metric than the final one marked on the graph, the RMS or root mean square. The RMS is the value required in all calculations and equations which allow electrical circuits to be analysed and their behaviour predicted. The reasons why this is the re-quired measurement relate to the amount of electrical power in the signal, but is not further addressed here. The name RMS itself is entirely self explanatory, it is calculated by taking the root of the mean of the square. In other words first square the signal, then calculate the mean (or average) of this, and finally take the square root of this 66 Electrical Theory 5.1 – Signal Wavelengths From Eq. 5.1 the relationship between the speed, frequency, and wavelength of a signal is known.

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  Engineering Circuit Analysis

  • J. David Irwin, R. Mark Nelms(Authors)
  • 2022(Publication Date)
  • Wiley

    (Publisher)

  It is common practice to specify the voltage rating of ac electrical devices in terms of the rms voltage. For example, if you examine an incandescent light bulb, you will see a voltage rating of 120 V, which is the RMS Value. For now, we will add an rms to our voltages and currents to indicate that we are using RMS Values in our calculations. Solution Substituting these expressions into Eq. (8.23) yields I rms = [ 1 _ T  0 T I 2 M cos 2 (ωt − θ) dt ] 1/2 Using the trigonometric identity cos 2 ϕ = 1 _ 2 + 1 _ 2 cos 2ϕ we find that the preceding equation can be expressed as I rms = I M { ω ___ 2π  0 2π/ω [ 1 _ 2 + 1 _ 2 cos (2ωt − 2θ) ] dt } 1/2 Since we know that the average or mean value of a cosine wave is zero, I rms = I M ( ω ___ 2π  0 2π/ω 1 __ 2 dt ) 1/2 = I M [ ω _ 2π ( t _ 2 )  0 2π/ω ] 1/2 = I M _ √ — 2 (8.24) Therefore, the RMS Value of a sinusoid is equal to the maximum value divided by √ — 2. Hence, a sinusoidal current with a maximum value of I M delivers the same average power to a resistor R as a dc current with a value of I M / √ — 2. Recall that earlier a phasor X was defined as X M / θ for a sinusoidal wave of the form X M cos (ωt + θ). This phasor can also be represented as X M / √ — 2 / θ if the units are given in rms. For example, 120 / 30° V rms is equivalent to 170 / 30° V. EXAMPLE 8.8 Determine the RMS Value of the current waveform in Fig. 8.9 and use this value to compute the average power delivered to a 6-Ω resistor through which this current is flowing. Solution The current waveform is periodic with a period of T = 4 s. The RMS Value is I rms = { 1 _ 4 [  0 2 (4) 2 dt +  2 4 (−4) 2 dt ]} 1/2 = [ 1 _ 4 ( 16t  0 2 + 16t  2 4 )] 1/2 = 4 A The average power delivered to a 6-Ω resistor with this current is P = I 2 rms R = (4) 2 (6) = 96 W 410 Chapter 8 Steady-State Power Analysis t (s) −2 4 −4 0 2 4 6 Current (A) i(t) FIGURE 8.9 Waveform used to illustrate RMS Values.

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