Understanding Alternating Current: Waveforms, Frequency, and Phases, Cheat Sheet of Programmable Logic Circuits

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Prepared by Md. Amirul Islam Lecturer Department of Applied Physics & Electronics Bangabandhu Sheikh Mujibur Rahman Science & Technology University, Gopalganj – 8100

Waveform: The path traced by a quantity, such as the voltage in Figure plotted as a function of some variable such as time, position, degrees, radians, temperature and so on. Figure: Sinusoidal Voltage Reference: Circuit Analysis by Robert Boylestad, Topic – 13.2, Page – 523 Instantaneous value: The magnitude of a waveform at any instant of time; denoted by lowercase letters (e 1 , e 2 ).

Peak amplitude: The maximum value of a waveform as measured from its average, or mean, value, denoted by uppercase letters (such as Em for sources of voltage and Vm for the voltage drop across a load). For the waveform of Figure, the average value is zero volts, and E m is as defined by the figure. Figure: Sinusoidal Voltage Reference: Circuit Analysis by Robert Boylestad, Topic – 13.2, Page – 523

Peak-to-peak value: Denoted by Ep-p or Vp-p, the full voltage between positive and negative peaks of the waveform, that is, the sum of the magnitude of the positive and negative peaks. Figure: Sinusoidal Voltage Reference: Circuit Analysis by Robert Boylestad, Topic – 13.2, Page – 523 Periodic waveform: A waveform that continually repeats itself after the same time interval. The waveform of figure is a periodic waveform.

Period (T ): The time interval between successive repetitions of a periodic waveform (the period T 1

= T

2

= T

3 in figure), as long as successive similar points of the periodic waveform are used in determining T. Figure: Sinusoidal Voltage Reference: Circuit Analysis by Robert Boylestad, Topic – 13.2, Page – 523

Frequency ( f ): The number of cycles that occur in 1 s. The frequency of the waveform of fig – (a) is 1 cycle per second, for fig – (b) is 2. 5 cycles per second. The unit of measure for frequency is the hertz (Hz), and 1Hz = 1 cycle per second Reference: Circuit Analysis by Robert Boylestad, Topic – 13.2, Page – 523 From definition we get,

Math. Problem: Find the period of a periodic waveform with a frequency of a. 60 Hz. b. 1000 Hz. Reference: Circuit Analysis by Robert Boylestad, Example – 13.1, Page – 527

Sinusoidal Wave: The sinusoidal waveform is the only alternating waveform whose shape is unaffected by the response characteristics of R, L, and C elements. Phase of the output wave may change. Reference: Circuit Analysis by Robert Boylestad, Topic– 13.5, Page – 535 General expression to represent a sinusoidal wave: v = Vm sin( ω t+ θ ) = Vm sin( 2 π ft+ θ )

Reference: - Math. Problem: Determine (a) peak value (b) peak to peak value (c) period (d) frequency (e) phase for the following sinusoidal waves: i. v = 60 sin( 1800 t+ 20 ° ) ii. v = 60 sin( 2 π 1000 t- 60 ° )

b. i = 15 sin( ωt + 60 ° ) and v = 10 sin( ωt – 20 ° ) Reference: Circuit Analysis by Robert Boylestad, Example– 13.12, Page – 537 Solution: From the figure – i leads v by 80 ° or v lags i by 80 ° c. i = 2 cos( ωt + 10 ° ) and v = 3 sin( ωt – 10 ° ) Solution: i = 2 cos(ωt+ 10 ° ) = 2 sin( ωt + 100 ° ) From the figure – i leads v by 110 ° or v lags I by 110 °

d. i = – sin( ωt + 30 ° ) and v = 2 sin( ωt + 10 ° ) Reference: Circuit Analysis by Robert Boylestad, Example – 13.12, Page – 537 Solution: i = – sin( ω t + 30 ° ) = sin( ω t – 150 ° ) = sin( ω t + 210 ° ) v leads i by 160 ° , or i lags v by 160 °. or, i leads v by 200 ° , or v lags i by 200 °.