Calculating Average and Root Mean Square (RMS) Values of Periodic Waves, Lecture notes of Engineering

Download Calculating Average and Root Mean Square (RMS) Values of Periodic Waves and more Lecture notes Engineering in PDF only on Docsity! Average and RMS Values WAVEFORM TERMS AND DEFINITIONS v Waveform – The variation of a quantity such as voltage on a graph as a function of time. v Cycle - The repetition of a variable recurring at equal intervals. v Period – The time taken for one cycle. v Instantaneous Value – The magnitude of a variable at any instant in time. v Peak-to-Peak Value – The maximum variation between maximum positive and maximum negative instantaneous values. v Frequency – The number of cycles occurring in one second. Relationship between Frequency and Period - T f 1 = AVERAGE VALUE A periodic function f(t) is shown in Figure 1. Note: A function is periodic if f(t) = f(t + nT) where n = integer and T = period Figure 1: Period function f(t) shows 12 values in a period of T The period T is broken up into 14 equal parts and at each interval of time, the value of the function is recorded. The average value of the function over the period T is given by Yav where: Yav = Yav1 + Yav2 Yav1 = Yav for n = 0 to n = 7 in tn Yav2 = Yav for n = 7 to n = 14 in tn 7 )()()()()()()( 7654321 1 tftftftftftftf Yav ++++++ = 7 )()()()()()()( 141312111098 2 tftftftftftftf Yav ++++++ = Area under curve from t0 to t7 = t{ f(t1) + f(t2) + f(t3) + f(t4) + f(t5) + f(t6) + f(t7) } { } t tftftftftftftf t 7 )()()()()()()( 7654321 ++++++= { } 2 )()()()()()()( 7654321 T tftftftftftftf t ++++++ = ∴ = Average value for interval Area under curve for one period = ∫ T dttf 0 ).( = ∫ T 0 f(t).dt T 1 ∴ Average value of f(t) = ∫ T 0 f(t).dt T 1 ∴ ∫= 2 T 0 av1 f(t).dtT/2 1 Y ∫= T 2 T av2 f(t).dtT/2 1 Y Area under curve from t0 to t7 Time Area under curve Time of interval Area under curve for one period Time of interval for one period