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The Study of Root Mean Square (RMS) Value Mechanical, Electrical, Electronics Engineering

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The Study of Root Mean Square (RMS) Value Mechanical, Electrical, Electronics Engineering The Study of Root Mean Square (RMS) Value Mechanical, Electrical, Electronics Engineering INTRODUCTION y So the RMS value RMS is given by: The root mean square value of a quantity is the T T square root of the mean value of the squared 1 1 y = ∫ y 2(t)dt = ∫ A2 sin2 ω tdt … (3) values of the quantity taken over an interval. The RMS T T 0 0 RMS value of any function y = f (t) over the range 2 t = a to t = b can be defined as: We know that cos2θ = 1− 2sin θ , hence: b 2 1 1 sin ω t = (1− cos2ω t ) … (4) RMS value = ∫ y 2dt 2 b − a a Substituting (4) into (3), we get, in sequence One of the principal applications of RMS values is T 1 A2 with alternating currents and voltages. y = ∫ (1− cos2ω t )dt RMS T 2 ROOT MEAN SQUARE (RMS) VALUE 0 The value of an AC voltage is continually T A2 changing from zero up to the positive peak, y = ∫ (1− cos2ω t ) dt RMS 2T through zero to the negative peak and back to 0 zero again. T A2 sin2ω t y = t − RMS ω 2T 2 0 A2 sin2ω T y = T − RMS 2T 2ω A2 A2 sin(2ω T ) y = − … (5) RMS 2 4ω T As T → ∞ , the second (oscillatory) term in equation (5) tends to zero. Hence, we obtain: Figure-1: Difference between peak and RMS voltage = A yRMS Clearly, for most of the time it is less than the 2 peak voltage, so this is not a good measure of its 1 real effect. Instead we use the root mean square ⇒ RMS Value of y = × Peak Value of y 2 V ≈ voltage ( RMS ) which is 1 2 0.7 of the peak ∴ = × yRMS 0.7 y peak voltage (Vpeak ): The RMS value is the effective value of a varying = × = × VRMS 0.7 Vpeak or Vpeak 1.4 VRMS voltage or current. It is the equivalent steady DC Similar equations also apply to the current. (constant) value which gives the same effect. For They are only true for sine waves (the most example, a lamp connected to a 6V RMS AC common type of AC) because the factors (here supply will shine with the same brightness when 0.7 and 1.4) take different values for other connected to a steady 6V DC supply. However, shapes. We can calculate these values as the lamp will be dimmer if connected to a 6V peak follows: AC supply because the RMS value of this is only Take a sine wave representing either current or 4.2V (it is equivalent to a steady 4.2V DC). voltage with peak value A: What do AC meters show? Is it the RMS or peak voltage? y(t) = Asinω t … (1) AC voltmeters and ammeters show the RMS ω = 2π f -1 where (rads ) and f = frequency (Hz). The value of the voltage or current. DC meters also time average y(T ) over period T (seconds) of the show the RMS value when connected to varying signal y(t) is given by: DC provided that the DC is varying quickly; if the T frequency is less than about 10Hz you will see the 1 y(T ) = ∫ y(t)dt … (2) meter reading fluctuating. T 0 What does '6V AC' really mean? Is it the RMS π 1 2 or peak voltage? RMS value = ∫ v 2dθ 2π − 0 If the peak value is meant it should be clearly 0 stated, otherwise assume it is the RMS value. In π 1 2 everyday use, AC voltages (and currents) are = ∫ (100 sinθ ) 2 dθ 2π always given as RMS values because this allows 0 a sensible comparison to be made with steady π DC voltages (and currents), such as from a 10000 2 = ∫ sin2 θ dθ battery. For example, a 6V AC supply means 6V 2π RMS with the peak voltage about 8.6V. The UK 0 mains supply is 230V AC; this means 230V RMS Using equation (4), we can write: so the peak voltage of the mains is about 320V! π 10000 2 1 So what does root mean square (RMS) really RMS value = ∫ (1− cos 2θ ) dθ 2π 2 mean? 0 First square all the values, then find the average 2π 10000 1 sin2θ (mean) of these square values over a complete = θ − π cycle, and finally find the square root of this 2 2 2 0 average. That is the RMS value. Confused? Then 10000 1 sin 4π sin0 just accept that RMS values for voltage and = 2π − − 0 − current are much more useful quantities than 2π 2 2 2 peak values. 10000 1 IMPORTANCE OF RMS VALUE = [2π ] 2π 2 Since a digital voltmeter measures the RMS value 100 of a fluctuating voltage, care must be taken when = = 70.71V finding the RMS values of very low frequency 2 signals otherwise inaccuracies can arise. The Please note: Being a sinusoidal wave, we can second term in equation (5) is an oscillation of calculate the RMS value for this voltage using the A2 relation derived on page-1, i.e. maximum amplitude that decreases with ω = × 4 T vRMS 0.7 v peak increasing period T. For a signal measured over a ⇒ v = 0.7 × 100 ≈ 70.71V time period T << ∞, there is obvious uncertainty RMS A But it is not so straight-forward in case of a non- over how much the true RMS deviates from . sinusoidal function as shown in the next worked 2 example. The maximum uncertainty is the maximum error over the assumed RMS and is given by: WORKED EXAMPLE – 2: 2 Find the RMS value for a voltage defined as A the following saw-tooth function over an ω 1 Maximum uncertainty = 4 T = … (6) interval [0, 2T]: A2 2ω T 2 So, at lower frequencies (smaller values of ω ), longer averaging times are required to reduce the uncertainty of the measurements to an acceptable level. Equivalent uncertainties may be derived for other non-sinusoidal waveforms. WORKED EXAMPLE – 1: V pk < < A sinusoidal voltage has a maximum value of t, for 0 t T 100V. Calculate its RMS value. V (t ) = T V Solution: A sinusoidal voltage v having a − + pk ( − ) < < Vpk t T , for T t 2T maximum value of 100V may be written as: T v (θ ) = 100 sinθ Solution: Since the function is symmetrical, we θ = θ = π can find the required RMS value by finding it only Over the range 0 to 2 (a complete cycle), the RMS value is given by: in the interval [0, T] as follows: 2 T V EXTENSION ACTIVITY – 2: = 1 pk VRMS ∫ t dt A company responsible for drying the timber used T − 0 T 0 in engineering applications has found that the 2 percentage moisture in timber, y, is related to the 1 V T = pk ∫ ( t ) 2dt number of days of drying, x, needed to become T 2 0.05 T 0 dry by the equation y ( x ) = − x − 0.25x + 100 . 2 T Determine, correct up to 3 significant figures, the V t 3 = pk RMS value of moisture present in the timber 3 3 between days x = 0 to x = 100 . Also plot a graph T 0 of this function using any available software and V 2 3 estimate after how many days, the timber will be = pk T completely dry, i.e. y = 0 . T 3 3 (Answer: 86.6%, Approximately 392 days ) Vpk = 3 EXTENSION ACTIVITY – 3: Hence, if the peak value of this voltage is 200V, The tidal height H at a particular location varies then its RMS value will be approximately 115V. with time t over a fortnight as: π t EXTENSION ACTIVITY – 1: H = (4 + 3.5cos )cos4π t A current, i ( t ) = 30 sin100π t amperes is applied 7 across an electric circuit. Determine its mean and where t is measured in days. To assess the RMS values, each correct to 4 significant figures, location’s potential as a site for generating over the range t = 0 to t = 10 milliseconds. electricity through tidal power, an engineer wants to know the RMS value of H over a fortnight, i.e. (Answers: 19.10 A, 21.21 A) ≤ ≤ for 0 t 14days . Plot H as a function of t and find its RMS value over this period. WHERE TO FIND MORE 1. Basic Engineering Mathematics, John Bird, 2007, published by Elsevier Ltd. 2. Engineering Mathematics, Fifth Edition, John Bird, 2007, published by Elsevier Ltd. 3. http://www.kpsec.freeuk.com/acdc.htm INFORMATION FOR TEACHERS The teachers should have some knowledge of . What RMS value is . Trigonometrical identities . Integration . Plotting graphs of any function using ICT TOPICS COVERED FROM “MATHEMATICS FOR ENGINEERING” . Topic 1: Mathematical Models in Engineering . Topic 3: Models of Oscillations . Topic 4: Functions . Topic 5: Geometry . Topic 6: Differentiation and Integration LEARNING OUTCOMES . LO 01: Understand the idea of mathematical modelling . LO 03: Understand the use of trigonometry to model situations involving oscillations . LO 04: Understand the mathematical structure of a range of functions and be familiar with their graphs . LO 06: Know how to use differentiation and integration in the context of engineering analysis and problem solving . LO 09: Construct rigorous mathematical arguments and proofs in engineering context . LO 10: Comprehend translations of common realistic engineering contexts into mathematics ASSESSMENT CRITERIA . AC 1.1: State assumptions made in establishing a specific mathematical model . AC 3.1: Solve problems in engineering requiring knowledge of trigonometric functions . AC 3.2: Relate trigonometrical expressions to situations involving oscillations .
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