Ac Theory
Alternating Current Theory - J R Lucas
An alternating waveform is a periodic waveform which alternate between positive and negative values. Unlike direct waveforms, they cannot be characterised by one magnitude as their amplitude is continuously varying from instant to instant. Thus various forms of magnitudes are defined for such waveforms. The advantage of the alternating waveform for electric power is that it can be stepped up or stepped down in potential easily for transmission and utilisation. Alternating waveforms can be of many shapes. The one that is used with electric power is the sinusoidal waveform. This has an equation of the form v(t) = Vm sin(ω t + φ )
If the period of the waveform is T, then its angular frequency ω corresponds to ωT = 2π. (a) Instantaneous value: The instantaneous value of a waveform is the value of the waveform at any given instant of time. It is a time variable a(t). For a sinusoid, Instantaneous value a(t) = Am sin(ω t+ φ ) (b) Peak value: The peak value, or maximum value, of a waveform is the maximum instantaneous value of the waveform. For a sinusoid, Peak value = Am (c) Mean value: The mean value of a waveform is equal to the mean value over a complete cycle of the waveform. It also corresponds to the direct component of the waveform. a(t) Amean −Am v(t) Vm φ/ω T t T 1 to +T Mean value Amean = ∫ a (t ).dt T to The mean value of a waveform which has equal positive and negative half cycles must thus be always zero. 1 t o +T For a sinusoid, Mean value = ∫ Am sin (ω t + φ ).dt = 0 T to (d) Average value (rectified): The full-wave rectified average value or average value of a waveform is defined as the mean value of the rectified waveform over a complete cycle. Average value Aavg For a sinusoid,
T T 1 2 Average value = ∫ Am sin ω t.dt − ∫ Am sin ω t.dt T 0 T 2 1 2 = ⋅ 2 Am = Am ωT π
Aavg
1 = ∫ a(t ).dt − ∫ a(t ).dt T positive negative hal fcycle hal fcycle vrect T T t The average
An alternating waveform is a periodic waveform which alternate between positive and negative values. Unlike direct waveforms, they cannot be characterised by one magnitude as their amplitude is continuously varying from instant to instant. Thus various forms of magnitudes are defined for such waveforms. The advantage of the alternating waveform for electric power is that it can be stepped up or stepped down in potential easily for transmission and utilisation. Alternating waveforms can be of many shapes. The one that is used with electric power is the sinusoidal waveform. This has an equation of the form v(t) = Vm sin(ω t + φ )
If the period of the waveform is T, then its angular frequency ω corresponds to ωT = 2π. (a) Instantaneous value: The instantaneous value of a waveform is the value of the waveform at any given instant of time. It is a time variable a(t). For a sinusoid, Instantaneous value a(t) = Am sin(ω t+ φ ) (b) Peak value: The peak value, or maximum value, of a waveform is the maximum instantaneous value of the waveform. For a sinusoid, Peak value = Am (c) Mean value: The mean value of a waveform is equal to the mean value over a complete cycle of the waveform. It also corresponds to the direct component of the waveform. a(t) Amean −Am v(t) Vm φ/ω T t T 1 to +T Mean value Amean = ∫ a (t ).dt T to The mean value of a waveform which has equal positive and negative half cycles must thus be always zero. 1 t o +T For a sinusoid, Mean value = ∫ Am sin (ω t + φ ).dt = 0 T to (d) Average value (rectified): The full-wave rectified average value or average value of a waveform is defined as the mean value of the rectified waveform over a complete cycle. Average value Aavg For a sinusoid,
T T 1 2 Average value = ∫ Am sin ω t.dt − ∫ Am sin ω t.dt T 0 T 2 1 2 = ⋅ 2 Am = Am ωT π
Aavg
1 = ∫ a(t ).dt − ∫ a(t ).dt T positive negative hal fcycle hal fcycle vrect T T t The average