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AC Fundamentals Radians Converting Between Radians and Degrees Frequency and Period Angular Velocity Generated Angle Instantaneous Current and Voltage Effective Value of ac Current or Voltage Capacitance and Inductance Radians Key Point: When the length of an arc equals the length of the radius of a circle, the angle is equal to 1 radian. One radian = 57.3°. Converting Between Radians and Degrees To convert from radians to degrees: To convert from radians to degrees: Convert 85o to radians Convert 3.42 radians to degrees angle = 37° Frequency and Period The frequency (f) is the number of times per second that we generate a sine wave. The period (T) is the time it takes to generate one complete sine wave. T = 400 μs. Find f f = 1/T T = 1/f f = 15 kHz. Find T Angular Velocity Angular velocity, ω, is the angular motion or speed of a rotating vector or phasor. Angular velocity is measured in radians per second or in degrees per second. ω = 2πf o ω = 360 f Generated Angle The generated angle at any instant, t, can be calculated as: angle in radians = 2πft = ωt angle in degrees = 360° ft = ωt Given: f = 2kHz t = 100 μs ωt = 1.26 rad ωt = 72.0° Self-Test 18-1 #4, 5, 6 Instantaneous Current and Voltage In ac circuits, instantaneous values of current and voltage may be found using the following equations: i = Ipk sin ωt e = Epk sin ωt Self-Test 18-2 #1, 2 Let Ipk = 40 mA, Epk = 12 V, and f = 5 kHz Find i and e after 70 μs i = 40 mApk sin ωt = 32.4 mAt e = 12 Vpk sin ωt = 9.71 Vt Effective Value of ac Current or Voltage The rms or effective value of an ac voltage or current is the value that converts the same energy as does a dc value. The rms value equals 0.707 of the maximum or peak of a sine wave. Irms = Ieff = 0.707 Ipk Vrms = Veff = 0.707 Vpk Ipk = 1/ Ieff = 1.414 Irms Vpk = 1/ Veff = 1.414 Vrms Capacitance and Inductance Key Point: Whenever we have capacitance or inductance in an ac circuit, the current and voltage are out of phase. When the phase angle, theta, θ, is known, we can calculate the instantaneous values of current and voltage. When current is the reference, i = Ipk sin ωt e = Epk sin (ωt ) When voltage is the reference, e = Epk sin ωt i = Ipk sin (ωt ) Self-Test 18-2 #5